What is the general solution for tan^2(x)+tan(x)-20=0 ?

The general solution for tan^2(x)+tan(x)-20=0 is x=1.32581…+pin,x=-1.37340…+pin

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tan^2(x)+tan(x)-20=0

Solution

tan^{2} (x) + tan (x) - 20 = 0

Solution

x = 1.32581 \dots + πn, x = - 1.37340 \dots + πn

Degrees

x = 75.96375 \dots^{\circ} + 18 0^{\circ} n, x = - 78.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + tan (x) - 20 = 0

Solve by substitution

tan^{2} (x) + tan (x) - 20 = 0

Let:

tan (x) = u

u^{2} + u - 20 = 0

u^{2} + u - 20 = 0 : u = 4, u = - 5

u^{2} + u - 20 = 0

Solve with the quadratic formula

u^{2} + u - 20 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 20

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 20) = 9

1^{2} - 4 \cdot 1 \cdot (- 20)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 20)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 20

Multiply the numbers:

4 \cdot 1 \cdot 20 = 80

= 1 + 80

Add the numbers:

1 + 80 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- 1 \pm 9}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 9}{2 \cdot 1}, u_{2} = \frac{- 1 - 9}{2 \cdot 1}

u = \frac{- 1 + 9}{2 \cdot 1} : 4

\frac{- 1 + 9}{2 \cdot 1}

Add/Subtract the numbers:

- 1 + 9 = 8

= \frac{8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- 1 - 9}{2 \cdot 1} : - 5

\frac{- 1 - 9}{2 \cdot 1}

Subtract the numbers:

- 1 - 9 = - 10

= \frac{- 10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10}{2}

Apply the fraction rule:

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

= - \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= - 5

The solutions to the quadratic equation are:

u = 4, u = - 5

Substitute back

u = tan (x)

tan (x) = 4, tan (x) = - 5

tan (x) = 4, tan (x) = - 5

tan (x) = 4 : x = arctan (4) + πn

tan (x) = 4

Apply trig inverse properties

tan (x) = 4

General solutions for

tan (x) = 4

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

x = arctan (4) + πn

x = arctan (4) + πn

tan (x) = - 5 : x = arctan (- 5) + πn

tan (x) = - 5

Apply trig inverse properties

tan (x) = - 5

General solutions for

tan (x) = - 5

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

x = arctan (- 5) + πn

x = arctan (- 5) + πn

Combine all the solutions

x = arctan (4) + πn, x = arctan (- 5) + πn

Show solutions in decimal form

x = 1.32581 \dots + πn, x = - 1.37340 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+tan(x)-20=0 ?
The general solution for tan^2(x)+tan(x)-20=0 is x=1.32581…+pin,x=-1.37340…+pin