What is the general solution for 3-cosh(x)=0 ?

The general solution for 3-cosh(x)=0 is x=ln(3-2sqrt(2)),x=ln(3+2sqrt(2))

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3-cosh(x)=0

Solution

3 - cosh (x) = 0

Solution

x = ln (3 - 22), x = ln (3 + 22)

Degrees

x = - 100.99797 \dots^{\circ}, x = 100.99797 \dots^{\circ}

Solution steps

3 - cosh (x) = 0

Rewrite using trig identities

3 - cosh (x) = 0

Use the Hyperbolic identity:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

3 - \frac{e ^{x} + e ^{- x}}{2} = 0

3 - \frac{e ^{x} + e ^{- x}}{2} = 0

3 - \frac{e ^{x} + e ^{- x}}{2} = 0 : x = ln (3 - 22), x = ln (3 + 22)

3 - \frac{e ^{x} + e ^{- x}}{2} = 0

Multiply both sides by

2

3 \cdot 2 - \frac{e ^{x} + e ^{- x}}{2} \cdot 2 = 0 \cdot 2

Simplify

6 - (e^{x} + e^{- x}) = 0

Add

(e^{x} + e^{- x})

to both sides

6 - (e^{x} + e^{- x}) + e^{x} + e^{- x} = 0 + e^{x} + e^{- x}

Simplify

6 - (e^{x} + e^{- x}) + e^{x} + e^{- x} = e^{x} + e^{- x}

Apply exponent rules

6 - (e^{x} + e^{- x}) + e^{x} + e^{- x} = e^{x} + e^{- x}

Apply exponent rule:

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

e^{- x} = (e^{x})^{- 1}

6 - (e^{x} + (e^{x})^{- 1}) + e^{x} + (e^{x})^{- 1} = e^{x} + (e^{x})^{- 1}

6 - (e^{x} + (e^{x})^{- 1}) + e^{x} + (e^{x})^{- 1} = e^{x} + (e^{x})^{- 1}

Rewrite the equation with

e^{x} = u

6 - (u + (u)^{- 1}) + u + (u)^{- 1} = u + (u)^{- 1}

Solve

6 - (u + u^{- 1}) + u + u^{- 1} = u + u^{- 1} : u = 3 - 22, u = 3 + 22

6 - (u + u^{- 1}) + u + u^{- 1} = u + u^{- 1}

Refine

6 - (u + \frac{1}{u}) + u + \frac{1}{u} = u + \frac{1}{u}

Subtract

u + \frac{1}{u}

from both sides

6 - (u + \frac{1}{u}) + u + \frac{1}{u} - (u + \frac{1}{u}) = u + \frac{1}{u} - (u + \frac{1}{u})

Simplify

6 - (u + \frac{1}{u}) = 0

Simplify

- (u + \frac{1}{u}) : - u - \frac{1}{u}

- (u + \frac{1}{u})

Distribute parentheses

= - (u) - (\frac{1}{u})

Apply minus-plus rules

+ (- a) = - a

= - u - \frac{1}{u}

6 - u - \frac{1}{u} = 0

Multiply both sides by

u

6 - u - \frac{1}{u} = 0

Multiply both sides by

u

6 u - uu - \frac{1}{u} u = 0 \cdot u

Simplify

6 u - uu - \frac{1}{u} u = 0 \cdot u

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

6 u - u^{2} - 1 = 0

6 u - u^{2} - 1 = 0

6 u - u^{2} - 1 = 0

Solve

6 u - u^{2} - 1 = 0 : u = 3 - 22, u = 3 + 22

6 u - u^{2} - 1 = 0

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} + 6 u - 1 = 0

Solve with the quadratic formula

- u^{2} + 6 u - 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = 6, c = - 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

6^{2} - 4 (- 1) (- 1) = 42

6^{2} - 4 (- 1) (- 1)

Apply rule

- (- a) = a

= 6^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

Subtract the numbers:

36 - 4 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

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

2

is a prime number, therefore no further factorization is possible

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

= 2^{5}

= 2^{5}

Apply exponent rule:

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

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

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

= 2^{2} 2

Refine

= 42

u_{1, 2} = \frac{- 6 \pm 4 2}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 6 + 4 2}{2 ( - 1 )}, u_{2} = \frac{- 6 - 4 2}{2 ( - 1 )}

u = \frac{- 6 + 4 2}{2 ( - 1 )} : 3 - 22

\frac{- 6 + 4 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 4 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 6 + 42 = - (6 - 42)

= \frac{6 - 4 2}{2}

Factor

6 - 42 : 2 (3 - 22)

6 - 42

Rewrite as

= 2 \cdot 3 - 2 \cdot 22

Factor out common term

2

= 2 (3 - 22)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 - 22

u = \frac{- 6 - 4 2}{2 ( - 1 )} : 3 + 22

\frac{- 6 - 4 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 4 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6 - 4 2}{- 2}

Apply the fraction rule:

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

- 6 - 42 = - (6 + 42)

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

Factor

6 + 42 : 2 (3 + 22)

6 + 42

Rewrite as

= 2 \cdot 3 + 2 \cdot 22

Factor out common term

2

= 2 (3 + 22)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 + 22

The solutions to the quadratic equation are:

u = 3 - 22, u = 3 + 22

u = 3 - 22, u = 3 + 22

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

6 - (u + u^{- 1}) + u + u^{- 1}

and compare to zero

u = 0

Take the denominator(s) of

u + u^{- 1}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3 - 22, u = 3 + 22

u = 3 - 22, u = 3 + 22

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 3 - 22 : x = ln (3 - 22)

e^{x} = 3 - 22

Apply exponent rules

e^{x} = 3 - 22

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (3 - 22)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3 - 22)

x = ln (3 - 22)

Solve

e^{x} = 3 + 22 : x = ln (3 + 22)

e^{x} = 3 + 22

Apply exponent rules

e^{x} = 3 + 22

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (3 + 22)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3 + 22)

x = ln (3 + 22)

x = ln (3 - 22), x = ln (3 + 22)

x = ln (3 - 22), x = ln (3 + 22)

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Frequently Asked Questions (FAQ)

What is the general solution for 3-cosh(x)=0 ?
The general solution for 3-cosh(x)=0 is x=ln(3-2sqrt(2)),x=ln(3+2sqrt(2))