What is the general solution for cos^4(x)+2sin^2(x)+6cos^2(x)+5=0 ?

The general solution for cos^4(x)+2sin^2(x)+6cos^2(x)+5=0 is No Solution for x\in\mathbb{R}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

cos^4(x)+2sin^2(x)+6cos^2(x)+5=0

Solution

cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x) + 5 = 0

Solution

No Solution for x \in R

Solution steps

cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x) + 5 = 0

Rewrite using trig identities

5 + cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x)

Simplify

5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x) : cos^{4} (x) + 4 cos^{2} (x) + 7

5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x)

Expand

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

2 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (x)

= 2 \cdot 1 - 2 cos^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (x)

= 5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x)

Simplify

5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x) : cos^{4} (x) + 4 cos^{2} (x) + 7

5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x)

Add similar elements:

- 2 cos^{2} (x) + 6 cos^{2} (x) = 4 cos^{2} (x)

= 5 + cos^{4} (x) + 2 + 4 cos^{2} (x)

Group like terms

= cos^{4} (x) + 4 cos^{2} (x) + 5 + 2

Add the numbers:

5 + 2 = 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

7 + cos^{4} (x) + 4 cos^{2} (x) = 0

Solve by substitution

7 + cos^{4} (x) + 4 cos^{2} (x) = 0

Let:

cos (x) = u

7 + u^{4} + 4 u^{2} = 0

7 + u^{4} + 4 u^{2} = 0 : u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

7 + u^{4} + 4 u^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} + 4 u^{2} + 7 = 0

Rewrite the equation with

a = u^{2}

and

a^{2} = u^{4}

a^{2} + 4 a + 7 = 0

Solve

a^{2} + 4 a + 7 = 0 : a = - 2 + 3 i, a = - 2 - 3 i

a^{2} + 4 a + 7 = 0

Solve with the quadratic formula

a^{2} + 4 a + 7 = 0

Quadratic Equation Formula:

For

a = 1, b = 4, c = 7

a_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 7}{2 \cdot 1}

a_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 7}{2 \cdot 1}

Simplify

4^{2} - 4 \cdot 1 \cdot 7 : 23 i

4^{2} - 4 \cdot 1 \cdot 7

Multiply the numbers:

4 \cdot 1 \cdot 7 = 28

= 4^{2} - 28

Apply imaginary number rule:

- a = i a

= i 28 - 4^{2}

- 4^{2} + 28 = 23

- 4^{2} + 28

4^{2} = 16

= - 16 + 28

Add/Subtract the numbers:

- 16 + 28 = 12

= 12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

= 3 2^{2}

Apply radical rule:

2^{2} = 2

= 23

= 23 i

a_{1, 2} = \frac{- 4 \pm 2 3 i}{2 \cdot 1}

Separate the solutions

a_{1} = \frac{- 4 + 2 3 i}{2 \cdot 1}, a_{2} = \frac{- 4 - 2 3 i}{2 \cdot 1}

a = \frac{- 4 + 2 3 i}{2 \cdot 1} : - 2 + 3 i

\frac{- 4 + 2 3 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4 + 2 3 i}{2}

Factor

- 4 + 23 i : 2 (- 2 + 3 i)

- 4 + 23 i

Rewrite as

= - 2 \cdot 2 + 23 i

Factor out common term

2

= 2 (- 2 + 3 i)

= \frac{2 ( - 2 + 3 i )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 2 + 3 i

a = \frac{- 4 - 2 3 i}{2 \cdot 1} : - 2 - 3 i

\frac{- 4 - 2 3 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4 - 2 3 i}{2}

Factor

- 4 - 23 i : - 2 (2 + 3 i)

- 4 - 23 i

Rewrite as

= - 2 \cdot 2 - 23 i

Factor out common term

2

= - 2 (2 + 3 i)

= - \frac{2 ( 2 + 3 i )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (2 + 3 i)

Negate

- (2 + 3 i) = - 2 - 3 i

= - 2 - 3 i

The solutions to the quadratic equation are:

a = - 2 + 3 i, a = - 2 - 3 i

a = - 2 + 3 i, a = - 2 - 3 i

Substitute back

a = u^{2},

solve for

u

Solve

u^{2} = - 2 + 3 i : u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

u^{2} = - 2 + 3 i

Substitute

u = a + bi

(a + bi)^{2} = - 2 + 3 i

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = - 2 + 3 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = - 2 2 ab = 3]

[a^{2} - b^{2} = - 2 2 ab = 3] : a = \frac{3}{2 2 + 7}, a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

[a^{2} - b^{2} = - 2 2 ab = 3]

Isolate

a

for

2 ab = 3 : a = \frac{3}{2 b}

2 ab = 3

Divide both sides by

2 b

2 ab = 3

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{3}{2 b}

Simplify

a = \frac{3}{2 b}

a = \frac{3}{2 b}

Plug the solutions

a = \frac{3}{2 b}

into

a^{2} - b^{2} = - 2

For

a^{2} - b^{2} = - 2

, subsitute

a

with

\frac{3}{2 b} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

For

a^{2} - b^{2} = - 2

, subsitute

a

with

\frac{3}{2 b}

(\frac{3}{2 b})^{2} - b^{2} = - 2

Solve

(\frac{3}{2 b})^{2} - b^{2} = - 2 : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

(\frac{3}{2 b})^{2} - b^{2} = - 2

Simplify

(\frac{3}{2 b})^{2} : \frac{3}{4 b ^{2}}

(\frac{3}{2 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 b)^{2} = 2^{2} b^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} b ^{2}}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= \frac{3}{2 ^{2} b ^{2}}

2^{2} = 4

= \frac{3}{4 b ^{2}}

\frac{3}{4 b ^{2}} - b^{2} = - 2

Multiply both sides by

4 b^{2}

\frac{3}{4 b ^{2}} - b^{2} = - 2

Multiply both sides by

4 b^{2}

\frac{3}{4 b ^{2}} \cdot 4 b^{2} - b^{2} \cdot 4 b^{2} = - 2 \cdot 4 b^{2}

Simplify

\frac{3}{4 b ^{2}} \cdot 4 b^{2} - b^{2} \cdot 4 b^{2} = - 2 \cdot 4 b^{2}

Simplify

\frac{3}{4 b ^{2}} \cdot 4 b^{2} : 3

\frac{3}{4 b ^{2}} \cdot 4 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 4 b ^{2}}{4 b ^{2}}

Cancel the common factor:

4

= \frac{3 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 3

Simplify

- b^{2} \cdot 4 b^{2} : - 4 b^{4}

- b^{2} \cdot 4 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 4 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 4 b^{4}

Simplify

- 2 \cdot 4 b^{2} : - 8 b^{2}

- 2 \cdot 4 b^{2}

Multiply the numbers:

2 \cdot 4 = 8

= - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

Solve

3 - 4 b^{4} = - 8 b^{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

3 - 4 b^{4} = - 8 b^{2}

Move

8 b^{2}

to the left side

3 - 4 b^{4} = - 8 b^{2}

Add

8 b^{2}

to both sides

3 - 4 b^{4} + 8 b^{2} = - 8 b^{2} + 8 b^{2}

Simplify

3 - 4 b^{4} + 8 b^{2} = 0

3 - 4 b^{4} + 8 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 4 b^{4} + 8 b^{2} + 3 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 4 u^{2} + 8 u + 3 = 0

Solve

- 4 u^{2} + 8 u + 3 = 0 : u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

- 4 u^{2} + 8 u + 3 = 0

Solve with the quadratic formula

- 4 u^{2} + 8 u + 3 = 0

Quadratic Equation Formula:

For

a = - 4, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

8^{2} - 4 (- 4) \cdot 3 = 47

8^{2} - 4 (- 4) \cdot 3

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 8^{2} + 48

8^{2} = 64

= 64 + 48

Add the numbers:

64 + 48 = 112

= 112

Prime factorization of

112 : 2^{4} \cdot 7

112

112

divides by

2112 = 56 \cdot 2

= 2 \cdot 56

56

divides by

256 = 28 \cdot 2

= 2 \cdot 2 \cdot 28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

= 2^{4} \cdot 7

= 2^{4} \cdot 7

Apply radical rule:

= 7 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 7

Refine

= 47

u_{1, 2} = \frac{- 8 \pm 4 7}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 8 + 4 7}{2 ( - 4 )}, u_{2} = \frac{- 8 - 4 7}{2 ( - 4 )}

u = \frac{- 8 + 4 7}{2 ( - 4 )} : - \frac{- 2 + 7}{2}

\frac{- 8 + 4 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 4 7}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8 + 4 7}{- 8}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 8 + 4 7}{8}

Cancel

\frac{- 8 + 4 7}{8} : \frac{7 - 2}{2}

\frac{- 8 + 4 7}{8}

Factor

- 8 + 47 : 4 (- 2 + 7)

- 8 + 47

Rewrite as

= - 4 \cdot 2 + 47

Factor out common term

4

= 4 (- 2 + 7)

= \frac{4 ( - 2 + 7 )}{8}

Cancel the common factor:

4

= \frac{- 2 + 7}{2}

= - \frac{7 - 2}{2}

= - \frac{- 2 + 7}{2}

u = \frac{- 8 - 4 7}{2 ( - 4 )} : \frac{2 + 7}{2}

\frac{- 8 - 4 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 4 7}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8 - 4 7}{- 8}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 8 - 47 = - (8 + 47)

= \frac{8 + 4 7}{8}

Factor

8 + 47 : 4 (2 + 7)

8 + 47

Rewrite as

= 4 \cdot 2 + 47

Factor out common term

4

= 4 (2 + 7)

= \frac{4 ( 2 + 7 )}{8}

Cancel the common factor:

4

= \frac{2 + 7}{2}

The solutions to the quadratic equation are:

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{- 2 + 7}{2} :

No Solution for

b \in R

b^{2} = - \frac{- 2 + 7}{2}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 + 7}{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b^{2} = \frac{2 + 7}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

The solutions are

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(\frac{3}{2 b})^{2} - b^{2}

and compare to zero

Solve

2 b = 0 : b = 0

2 b = 0

Divide both sides by

2

2 b = 0

Divide both sides by

2

\frac{2 b}{2} = \frac{0}{2}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

Plug the solutions

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

into

2 ab = 3

For

2 ab = 3

, subsitute

b

with

\frac{2 + 7}{2} : a = \frac{3}{2 2 + 7}

For

2 ab = 3

, subsitute

b

with

\frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = 3

Solve

2 a \frac{2 + 7}{2} = 3 : a = \frac{3}{2 2 + 7}

2 a \frac{2 + 7}{2} = 3

Divide both sides by

2 \frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = 3

Divide both sides by

2 \frac{2 + 7}{2}

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{3}{2 \frac{2 + 7}{2}}

Simplify

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{3}{2 \frac{2 + 7}{2}}

Simplify

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} : a

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

Divide the numbers:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

Cancel the common factor:

\frac{2 + 7}{2}

= a

Simplify

\frac{3}{2 \frac{2 + 7}{2}} : \frac{3}{2 2 + 7}

\frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= \frac{3}{2 \cdot \frac{2 + 7}{2}}

Multiply

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 + 7 \cdot 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} 2 + 7

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 2 + 7

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2 2 + 7

= \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

For

2 ab = 3

, subsitute

b

with

- \frac{2 + 7}{2} : a = - \frac{3}{2 2 + 7}

For

2 ab = 3

, subsitute

b

with

- \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = 3

Solve

2 a - \frac{2 + 7}{2} = 3 : a = - \frac{3}{2 2 + 7}

2 a - \frac{2 + 7}{2} = 3

Divide both sides by

2 - \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = 3

Divide both sides by

2 - \frac{2 + 7}{2}

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{3}{2 ( - \frac{2 + 7}{2} )}

Simplify

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{3}{2 ( - \frac{2 + 7}{2} )}

Simplify

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} : a

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )}

Remove parentheses:

(- a) = - a

= \frac{- 2 a \frac{2 + 7}{2}}{- 2 \frac{2 + 7}{2}}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

Divide the numbers:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

Cancel the common factor:

\frac{2 + 7}{2}

= a

Simplify

\frac{3}{2 ( - \frac{2 + 7}{2} )} : - \frac{3}{2 2 + 7}

\frac{3}{2 ( - \frac{2 + 7}{2} )}

Remove parentheses:

(- a) = - a

= \frac{3}{- 2 \frac{2 + 7}{2}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= - \frac{3}{2 \cdot \frac{2 + 7}{2}}

Multiply

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 + 7 \cdot 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} 2 + 7

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 2 + 7

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2 2 + 7

= - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = - 2

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2} :

True

a^{2} - b^{2} = - 2

Plug in

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

(- \frac{3}{2 2 + 7})^{2} - - \frac{2 + 7}{2}^{2} = - 2

Refine

- 2 = - 2

True

Check the solution

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} :

True

a^{2} - b^{2} = - 2

Plug in

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

(\frac{3}{2 2 + 7})^{2} - \frac{2 + 7}{2}^{2} = - 2

Refine

- 2 = - 2

True

Check the solutions by plugging them into

2 ab = 3

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2} :

True

2 ab = 3

Plug in

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

2 (- \frac{3}{2 2 + 7}) - \frac{2 + 7}{2} = 3

Refine

3 = 3

True

Check the solution

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} :

True

2 ab = 3

Plug in

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

2 \cdot \frac{3}{2 2 + 7} \frac{2 + 7}{2} = 3

Refine

3 = 3

True

Therefore, the final solutions for

a^{2} - b^{2} = - 2, 2 ab = 3

are

a = \frac{3}{2 2 + 7}, a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

Substitute back

u = a + bi

u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

Solve

u^{2} = - 2 - 3 i : u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

u^{2} = - 2 - 3 i

Substitute

u = a + bi

(a + bi)^{2} = - 2 - 3 i

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = - 2 - 3 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = - 2 2 ab = - 3]

[a^{2} - b^{2} = - 2 2 ab = - 3] : a = - \frac{3}{2 2 + 7}, a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

[a^{2} - b^{2} = - 2 2 ab = - 3]

Isolate

a

for

2 ab = - 3 : a = - \frac{3}{2 b}

2 ab = - 3

Divide both sides by

2 b

2 ab = - 3

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{- 3}{2 b}

Simplify

a = - \frac{3}{2 b}

a = - \frac{3}{2 b}

Plug the solutions

a = - \frac{3}{2 b}

into

a^{2} - b^{2} = - 2

For

a^{2} - b^{2} = - 2

, subsitute

a

with

- \frac{3}{2 b} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

For

a^{2} - b^{2} = - 2

, subsitute

a

with

- \frac{3}{2 b}

(- \frac{3}{2 b})^{2} - b^{2} = - 2

Solve

(- \frac{3}{2 b})^{2} - b^{2} = - 2 : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

(- \frac{3}{2 b})^{2} - b^{2} = - 2

Simplify

(- \frac{3}{2 b})^{2} : \frac{3}{4 b ^{2}}

(- \frac{3}{2 b})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{3}{2 b})^{2} = (\frac{3}{2 b})^{2}

= (\frac{3}{2 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 b)^{2} = 2^{2} b^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} b ^{2}}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= \frac{3}{2 ^{2} b ^{2}}

2^{2} = 4

= \frac{3}{4 b ^{2}}

\frac{3}{4 b ^{2}} - b^{2} = - 2

Multiply both sides by

4 b^{2}

\frac{3}{4 b ^{2}} - b^{2} = - 2

Multiply both sides by

4 b^{2}

\frac{3}{4 b ^{2}} \cdot 4 b^{2} - b^{2} \cdot 4 b^{2} = - 2 \cdot 4 b^{2}

Simplify

\frac{3}{4 b ^{2}} \cdot 4 b^{2} - b^{2} \cdot 4 b^{2} = - 2 \cdot 4 b^{2}

Simplify

\frac{3}{4 b ^{2}} \cdot 4 b^{2} : 3

\frac{3}{4 b ^{2}} \cdot 4 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 4 b ^{2}}{4 b ^{2}}

Cancel the common factor:

4

= \frac{3 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 3

Simplify

- b^{2} \cdot 4 b^{2} : - 4 b^{4}

- b^{2} \cdot 4 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 4 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 4 b^{4}

Simplify

- 2 \cdot 4 b^{2} : - 8 b^{2}

- 2 \cdot 4 b^{2}

Multiply the numbers:

2 \cdot 4 = 8

= - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

Solve

3 - 4 b^{4} = - 8 b^{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

3 - 4 b^{4} = - 8 b^{2}

Move

8 b^{2}

to the left side

3 - 4 b^{4} = - 8 b^{2}

Add

8 b^{2}

to both sides

3 - 4 b^{4} + 8 b^{2} = - 8 b^{2} + 8 b^{2}

Simplify

3 - 4 b^{4} + 8 b^{2} = 0

3 - 4 b^{4} + 8 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 4 b^{4} + 8 b^{2} + 3 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 4 u^{2} + 8 u + 3 = 0

Solve

- 4 u^{2} + 8 u + 3 = 0 : u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

- 4 u^{2} + 8 u + 3 = 0

Solve with the quadratic formula

- 4 u^{2} + 8 u + 3 = 0

Quadratic Equation Formula:

For

a = - 4, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

8^{2} - 4 (- 4) \cdot 3 = 47

8^{2} - 4 (- 4) \cdot 3

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 8^{2} + 48

8^{2} = 64

= 64 + 48

Add the numbers:

64 + 48 = 112

= 112

Prime factorization of

112 : 2^{4} \cdot 7

112

112

divides by

2112 = 56 \cdot 2

= 2 \cdot 56

56

divides by

256 = 28 \cdot 2

= 2 \cdot 2 \cdot 28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

= 2^{4} \cdot 7

= 2^{4} \cdot 7

Apply radical rule:

= 7 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 7

Refine

= 47

u_{1, 2} = \frac{- 8 \pm 4 7}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 8 + 4 7}{2 ( - 4 )}, u_{2} = \frac{- 8 - 4 7}{2 ( - 4 )}

u = \frac{- 8 + 4 7}{2 ( - 4 )} : - \frac{- 2 + 7}{2}

\frac{- 8 + 4 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 4 7}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8 + 4 7}{- 8}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 8 + 4 7}{8}

Cancel

\frac{- 8 + 4 7}{8} : \frac{7 - 2}{2}

\frac{- 8 + 4 7}{8}

Factor

- 8 + 47 : 4 (- 2 + 7)

- 8 + 47

Rewrite as

= - 4 \cdot 2 + 47

Factor out common term

4

= 4 (- 2 + 7)

= \frac{4 ( - 2 + 7 )}{8}

Cancel the common factor:

4

= \frac{- 2 + 7}{2}

= - \frac{7 - 2}{2}

= - \frac{- 2 + 7}{2}

u = \frac{- 8 - 4 7}{2 ( - 4 )} : \frac{2 + 7}{2}

\frac{- 8 - 4 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 4 7}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 8 - 4 7}{- 8}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 8 - 47 = - (8 + 47)

= \frac{8 + 4 7}{8}

Factor

8 + 47 : 4 (2 + 7)

8 + 47

Rewrite as

= 4 \cdot 2 + 47

Factor out common term

4

= 4 (2 + 7)

= \frac{4 ( 2 + 7 )}{8}

Cancel the common factor:

4

= \frac{2 + 7}{2}

The solutions to the quadratic equation are:

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{- 2 + 7}{2} :

No Solution for

b \in R

b^{2} = - \frac{- 2 + 7}{2}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 + 7}{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b^{2} = \frac{2 + 7}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

The solutions are

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(- \frac{3}{2 b})^{2} - b^{2}

and compare to zero

Solve

2 b = 0 : b = 0

2 b = 0

Divide both sides by

2

2 b = 0

Divide both sides by

2

\frac{2 b}{2} = \frac{0}{2}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

Plug the solutions

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

into

2 ab = - 3

For

2 ab = - 3

, subsitute

b

with

\frac{2 + 7}{2} : a = - \frac{3}{2 2 + 7}

For

2 ab = - 3

, subsitute

b

with

\frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = - 3

Solve

2 a \frac{2 + 7}{2} = - 3 : a = - \frac{3}{2 2 + 7}

2 a \frac{2 + 7}{2} = - 3

Divide both sides by

2 \frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = - 3

Divide both sides by

2 \frac{2 + 7}{2}

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{- 3}{2 \frac{2 + 7}{2}}

Simplify

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{- 3}{2 \frac{2 + 7}{2}}

Simplify

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} : a

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

Divide the numbers:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

Cancel the common factor:

\frac{2 + 7}{2}

= a

Simplify

\frac{- 3}{2 \frac{2 + 7}{2}} : - \frac{3}{2 2 + 7}

\frac{- 3}{2 \frac{2 + 7}{2}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= - \frac{3}{2 \cdot \frac{2 + 7}{2}}

Multiply

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 + 7 \cdot 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} 2 + 7

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 2 + 7

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2 2 + 7

= - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

For

2 ab = - 3

, subsitute

b

with

- \frac{2 + 7}{2} : a = \frac{3}{2 2 + 7}

For

2 ab = - 3

, subsitute

b

with

- \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = - 3

Solve

2 a - \frac{2 + 7}{2} = - 3 : a = \frac{3}{2 2 + 7}

2 a - \frac{2 + 7}{2} = - 3

Divide both sides by

2 - \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = - 3

Divide both sides by

2 - \frac{2 + 7}{2}

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{- 3}{2 ( - \frac{2 + 7}{2} )}

Simplify

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{- 3}{2 ( - \frac{2 + 7}{2} )}

Simplify

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} : a

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )}

Remove parentheses:

(- a) = - a

= \frac{- 2 a \frac{2 + 7}{2}}{- 2 \frac{2 + 7}{2}}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

Divide the numbers:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

Cancel the common factor:

\frac{2 + 7}{2}

= a

Simplify

\frac{- 3}{2 ( - \frac{2 + 7}{2} )} : \frac{3}{2 2 + 7}

\frac{- 3}{2 ( - \frac{2 + 7}{2} )}

Remove parentheses:

(- a) = - a

= \frac{- 3}{- 2 \frac{2 + 7}{2}}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= \frac{3}{2 \cdot \frac{2 + 7}{2}}

Multiply

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 + 7 \cdot 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} 2 + 7

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 2 + 7

Apply radical rule:

2^{\frac{1}{2}} = 2

= 2 2 + 7

= \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = - 2

Remove the ones that don't agree with the equation.

Check the solution

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2} :

True

a^{2} - b^{2} = - 2

Plug in

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

(\frac{3}{2 2 + 7})^{2} - - \frac{2 + 7}{2}^{2} = - 2

Refine

- 2 = - 2

True

Check the solution

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} :

True

a^{2} - b^{2} = - 2

Plug in

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

(- \frac{3}{2 2 + 7})^{2} - \frac{2 + 7}{2}^{2} = - 2

Refine

- 2 = - 2

True

Check the solutions by plugging them into

2 ab = - 3

Remove the ones that don't agree with the equation.

Check the solution

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2} :

True

2 ab = - 3

Plug in

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

2 \cdot \frac{3}{2 2 + 7} - \frac{2 + 7}{2} = - 3

Refine

- 3 = - 3

True

Check the solution

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} :

True

2 ab = - 3

Plug in

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

2 (- \frac{3}{2 2 + 7}) \frac{2 + 7}{2} = - 3

Refine

- 3 = - 3

True

Therefore, the final solutions for

a^{2} - b^{2} = - 2, 2 ab = - 3

are

a = - \frac{3}{2 2 + 7}, a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

Substitute back

u = a + bi

u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

The solutions are

u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

Substitute back

u = cos (x)

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i :

No Solution

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

Simplify

\frac{3}{2 2 + 7} + \frac{2 + 7}{2} i : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3} + i \frac{2 + 7}{2}

\frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

Multiply by the conjugate

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

Apply radical rule:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

Multiply by the conjugate

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 4, b = 27

= 4^{2} - (27)^{2}

Simplify

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (7)^{2}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

Multiply the numbers:

4 \cdot 7 = 28

= 28

= 16 - 28

Subtract the numbers:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Cancel

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Factor

4 - 27 : 2 (2 - 7)

4 - 27

Rewrite as

= 2 \cdot 2 - 27

Factor out common term

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

Cancel the common factor:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6} + i \frac{2 + 7}{2}

Rewrite

- \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

in standard complex form:

\frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

- \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

Apply radical rule:

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

Apply radical rule:

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

Expand

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

Apply the distributive law:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

Apply radical rule:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

Expand

7 (2 + 7) : 14 + 77

7 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

Multiply the numbers:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= - \frac{2 2 + 7 - 14 + 7 7}{6} + i \frac{2 + 7}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 2 + 7 - 14 + 7 7}{6} = - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6})

= - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6}) + i \frac{2 + 7}{2}

Remove parentheses:

(a) = a, - (- a) = a

= - \frac{2 2 + 7}{6} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Cancel

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 + 7}{2 3}

Cancel

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Group the real part and the imaginary part of the complex number

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + \frac{2 + 7}{2} i

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} = \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6}

Convert element to fraction:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= - \frac{2 2 + 7}{3} + \frac{\frac{14 + 7 7}{6} 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

Apply radical rule:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 14 + 7 7 ) \cdot 3}{6}

Cancel the common factor:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

Rationalize

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3} : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{( - 2 2 + 7 + \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

No Solution

cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i :

No Solution

cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

Simplify

- \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i : \frac{3 ( - \frac{14 + 7 7}{2} + 2 2 + 7 )}{3} - i \frac{2 + 7}{2}

- \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

Multiply by the conjugate

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

Apply radical rule:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

Multiply by the conjugate

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 4, b = 27

= 4^{2} - (27)^{2}

Simplify

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (7)^{2}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

Multiply the numbers:

4 \cdot 7 = 28

= 28

= 16 - 28

Subtract the numbers:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Cancel

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Factor

4 - 27 : 2 (2 - 7)

4 - 27

Rewrite as

= 2 \cdot 2 - 27

Factor out common term

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

Cancel the common factor:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - i \frac{2 + 7}{2}

Rewrite

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - \frac{2 + 7}{2} i

in standard complex form:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - \frac{2 + 7}{2} i

Apply rule

- (- a) = a

= \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

Apply radical rule:

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

Apply radical rule:

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

Expand

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

Apply the distributive law:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

Apply radical rule:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

Expand

7 (2 + 7) : 14 + 77

7 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

Multiply the numbers:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= \frac{2 2 + 7 - 14 + 7 7}{6} - i \frac{2 + 7}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 2 + 7 - 14 + 7 7}{6} = \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6}

= \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Cancel

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 + 7}{2 3}

Cancel

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Group the real part and the imaginary part of the complex number

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - \frac{2 + 7}{2} i

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} = \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6}

Convert element to fraction:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= \frac{2 2 + 7}{3} - \frac{\frac{14 + 7 7}{6} 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 2 + 7 - \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

Apply radical rule:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 14 + 7 7 ) \cdot 3}{6}

Cancel the common factor:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

Rationalize

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3} : \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{( 2 2 + 7 - \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

No Solution

cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i :

No Solution

cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

Simplify

- \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i : \frac{3 ( - \frac{14 + 7 7}{2} + 2 2 + 7 )}{3} + i \frac{2 + 7}{2}

- \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

Multiply by the conjugate

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

Apply radical rule:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

Multiply by the conjugate

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 4, b = 27

= 4^{2} - (27)^{2}

Simplify

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (7)^{2}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

Multiply the numbers:

4 \cdot 7 = 28

= 28

= 16 - 28

Subtract the numbers:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Cancel

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Factor

4 - 27 : 2 (2 - 7)

4 - 27

Rewrite as

= 2 \cdot 2 - 27

Factor out common term

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

Cancel the common factor:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + i \frac{2 + 7}{2}

Rewrite

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + \frac{2 + 7}{2} i

in standard complex form:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + \frac{2 + 7}{2} i

Apply rule

- (- a) = a

= \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

Apply radical rule:

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

Apply radical rule:

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

Expand

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

Apply the distributive law:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

Apply radical rule:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

Expand

7 (2 + 7) : 14 + 77

7 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

Multiply the numbers:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= \frac{2 2 + 7 - 14 + 7 7}{6} + i \frac{2 + 7}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 2 + 7 - 14 + 7 7}{6} = \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6}

= \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Cancel

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 + 7}{2 3}

Cancel

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

Group the real part and the imaginary part of the complex number

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + \frac{2 + 7}{2} i

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} = \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6}

Convert element to fraction:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= \frac{2 2 + 7}{3} - \frac{\frac{14 + 7 7}{6} 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 2 + 7 - \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

Apply radical rule:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 14 + 7 7 ) \cdot 3}{6}

Cancel the common factor:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

Rationalize

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3} : \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{( 2 2 + 7 - \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

No Solution

cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i :

No Solution

cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

Simplify

\frac{3}{2 2 + 7} - \frac{2 + 7}{2} i : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3} - i \frac{2 + 7}{2}

\frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

Apply radical rule:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

Multiply by the conjugate

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

Apply radical rule:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

Multiply by the conjugate

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 4, b = 27

= 4^{2} - (27)^{2}

Simplify

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (7)^{2}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

Multiply the numbers:

4 \cdot 7 = 28

= 28

= 16 - 28

Subtract the numbers:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Cancel

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

Factor

4 - 27 : 2 (2 - 7)

4 - 27

Rewrite as

= 2 \cdot 2 - 27

Factor out common term

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

Cancel the common factor:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6} - i \frac{2 + 7}{2}

Rewrite

- \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

in standard complex form:

\frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

- \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

Apply radical rule:

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

Apply radical rule:

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

Expand

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

Apply the distributive law:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

Apply radical rule:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

Expand

7 (2 + 7) : 14 + 77

7 (2 + 7)

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

Multiply the numbers:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= - \frac{2 2 + 7 - 14 + 7 7}{6} - i \frac{2 + 7}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 2 + 7 - 14 + 7 7}{6} = - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6})

= - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6}) - i \frac{2 + 7}{2}

Remove parentheses:

(a) = a, - (- a) = a

= - \frac{2 2 + 7}{6} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Cancel

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 + 7}{2 3}

Cancel

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

Group the real part and the imaginary part of the complex number

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - \frac{2 + 7}{2} i

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} = \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6}

Convert element to fraction:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= - \frac{2 2 + 7}{3} + \frac{\frac{14 + 7 7}{6} 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

Apply radical rule:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 14 + 7 7 ) \cdot 3}{6}

Cancel the common factor:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

Rationalize

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3} : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{( - 2 2 + 7 + \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

No Solution

Combine all the solutions

No Solution for x \in R

Graph

Popular Examples

1+sin(2a)=sin^2(a)

((cos^3(a)))/((2cos^2(a)-1))=cos(a)

cos(x-45)=0

7tan^2(x)-15=0

1+cos^2(x)-2cos^2(x/2)=0

Frequently Asked Questions (FAQ)

What is the general solution for cos^4(x)+2sin^2(x)+6cos^2(x)+5=0 ?
The general solution for cos^4(x)+2sin^2(x)+6cos^2(x)+5=0 is No Solution for x\in\mathbb{R}