What is the general solution for tan^2(x)+tan(x)-12=2 ?

The general solution for tan^2(x)+tan(x)-12=2 is x=1.27443…+pin,x=-1.34100…+pin

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Popular Trigonometry >

tan^2(x)+tan(x)-12=2

Solution

tan^{2} (x) + tan (x) - 12 = 2

Solution

x = 1.27443 \dots + πn, x = - 1.34100 \dots + πn

Degrees

x = 73.01988 \dots^{\circ} + 18 0^{\circ} n, x = - 76.83395 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + tan (x) - 12 = 2

Solve by substitution

tan^{2} (x) + tan (x) - 12 = 2

Let:

tan (x) = u

u^{2} + u - 12 = 2

u^{2} + u - 12 = 2 : u = \frac{- 1 + 57}{2}, u = \frac{- 1 - 57}{2}

u^{2} + u - 12 = 2

Move

2

to the left side

u^{2} + u - 12 = 2

Subtract

2

from both sides

u^{2} + u - 12 - 2 = 2 - 2

Simplify

u^{2} + u - 14 = 0

u^{2} + u - 14 = 0

Solve with the quadratic formula

u^{2} + u - 14 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 14

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 14 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 14 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 14) = 57

1^{2} - 4 \cdot 1 \cdot (- 14)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 14)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 14

Multiply the numbers:

4 \cdot 1 \cdot 14 = 56

= 1 + 56

Add the numbers:

1 + 56 = 57

= 57

u_{1, 2} = \frac{- 1 \pm 57}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 57}{2 \cdot 1}, u_{2} = \frac{- 1 - 57}{2 \cdot 1}

u = \frac{- 1 + 57}{2 \cdot 1} : \frac{- 1 + 57}{2}

\frac{- 1 + 57}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + 57}{2}

u = \frac{- 1 - 57}{2 \cdot 1} : \frac{- 1 - 57}{2}

\frac{- 1 - 57}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - 57}{2}

The solutions to the quadratic equation are:

u = \frac{- 1 + 57}{2}, u = \frac{- 1 - 57}{2}

Substitute back

u = tan (x)

tan (x) = \frac{- 1 + 57}{2}, tan (x) = \frac{- 1 - 57}{2}

tan (x) = \frac{- 1 + 57}{2}, tan (x) = \frac{- 1 - 57}{2}

tan (x) = \frac{- 1 + 57}{2} : x = arctan (\frac{- 1 + 57}{2}) + πn

tan (x) = \frac{- 1 + 57}{2}

Apply trig inverse properties

tan (x) = \frac{- 1 + 57}{2}

General solutions for

tan (x) = \frac{- 1 + 57}{2}

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

x = arctan (\frac{- 1 + 57}{2}) + πn

x = arctan (\frac{- 1 + 57}{2}) + πn

tan (x) = \frac{- 1 - 57}{2} : x = arctan (\frac{- 1 - 57}{2}) + πn

tan (x) = \frac{- 1 - 57}{2}

Apply trig inverse properties

tan (x) = \frac{- 1 - 57}{2}

General solutions for

tan (x) = \frac{- 1 - 57}{2}

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

x = arctan (\frac{- 1 - 57}{2}) + πn

x = arctan (\frac{- 1 - 57}{2}) + πn

Combine all the solutions

x = arctan (\frac{- 1 + 57}{2}) + πn, x = arctan (\frac{- 1 - 57}{2}) + πn

Show solutions in decimal form

x = 1.27443 \dots + πn, x = - 1.34100 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+tan(x)-12=2 ?
The general solution for tan^2(x)+tan(x)-12=2 is x=1.27443…+pin,x=-1.34100…+pin