What is the general solution for 4tan^2(x)+21tan(x)-49=0 ?

The general solution for 4tan^2(x)+21tan(x)-49=0 is x=1.05165…+pin,x=-1.42889…+pin

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4tan^2(x)+21tan(x)-49=0

Solution

4 tan^{2} (x) + 21 tan (x) - 49 = 0

Solution

x = 1.05165 \dots + πn, x = - 1.42889 \dots + πn

Degrees

x = 60.25511 \dots^{\circ} + 18 0^{\circ} n, x = - 81.86989 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) + 21 tan (x) - 49 = 0

Solve by substitution

4 tan^{2} (x) + 21 tan (x) - 49 = 0

Let:

tan (x) = u

4 u^{2} + 21 u - 49 = 0

4 u^{2} + 21 u - 49 = 0 : u = \frac{7}{4}, u = - 7

4 u^{2} + 21 u - 49 = 0

Solve with the quadratic formula

4 u^{2} + 21 u - 49 = 0

Quadratic Equation Formula:

For

a = 4, b = 21, c = - 49

u_{1, 2} = \frac{- 21 \pm 2 1 ^{2} - 4 \cdot 4 ( - 49 )}{2 \cdot 4}

u_{1, 2} = \frac{- 21 \pm 2 1 ^{2} - 4 \cdot 4 ( - 49 )}{2 \cdot 4}

2 1^{2} - 4 \cdot 4 (- 49) = 35

2 1^{2} - 4 \cdot 4 (- 49)

Apply rule

- (- a) = a

= 2 1^{2} + 4 \cdot 4 \cdot 49

Multiply the numbers:

4 \cdot 4 \cdot 49 = 784

= 2 1^{2} + 784

2 1^{2} = 441

= 441 + 784

Add the numbers:

441 + 784 = 1225

= 1225

Factor the number:

1225 = 3 5^{2}

= 3 5^{2}

Apply radical rule:

3 5^{2} = 35

= 35

u_{1, 2} = \frac{- 21 \pm 35}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 21 + 35}{2 \cdot 4}, u_{2} = \frac{- 21 - 35}{2 \cdot 4}

u = \frac{- 21 + 35}{2 \cdot 4} : \frac{7}{4}

\frac{- 21 + 35}{2 \cdot 4}

Add/Subtract the numbers:

- 21 + 35 = 14

= \frac{14}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{14}{8}

Cancel the common factor:

2

= \frac{7}{4}

u = \frac{- 21 - 35}{2 \cdot 4} : - 7

\frac{- 21 - 35}{2 \cdot 4}

Subtract the numbers:

- 21 - 35 = - 56

= \frac{- 56}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 56}{8}

Apply the fraction rule:

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

= - \frac{56}{8}

Divide the numbers:

\frac{56}{8} = 7

= - 7

The solutions to the quadratic equation are:

u = \frac{7}{4}, u = - 7

Substitute back

u = tan (x)

tan (x) = \frac{7}{4}, tan (x) = - 7

tan (x) = \frac{7}{4}, tan (x) = - 7

tan (x) = \frac{7}{4} : x = arctan (\frac{7}{4}) + πn

tan (x) = \frac{7}{4}

Apply trig inverse properties

tan (x) = \frac{7}{4}

General solutions for

tan (x) = \frac{7}{4}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{7}{4}) + πn

x = arctan (\frac{7}{4}) + πn

tan (x) = - 7 : x = arctan (- 7) + πn

tan (x) = - 7

Apply trig inverse properties

tan (x) = - 7

General solutions for

tan (x) = - 7

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 7) + πn

x = arctan (- 7) + πn

Combine all the solutions

x = arctan (\frac{7}{4}) + πn, x = arctan (- 7) + πn

Show solutions in decimal form

x = 1.05165 \dots + πn, x = - 1.42889 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)+21tan(x)-49=0 ?
The general solution for 4tan^2(x)+21tan(x)-49=0 is x=1.05165…+pin,x=-1.42889…+pin