What is the general solution for tan^2(x)+tan(x)-6=0 ?

The general solution for tan^2(x)+tan(x)-6=0 is x=1.10714…+pin,x=-1.24904…+pin

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tan^2(x)+tan(x)-6=0

Solution

tan^{2} (x) + tan (x) - 6 = 0

Solution

x = 1.10714 \dots + πn, x = - 1.24904 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + tan (x) - 6 = 0

Solve by substitution

tan^{2} (x) + tan (x) - 6 = 0

Let:

tan (x) = u

u^{2} + u - 6 = 0

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

u^{2} + u - 6 = 0

Solve with the quadratic formula

u^{2} + u - 6 = 0

Quadratic Equation Formula:

For

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

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

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

1^{2} - 4 \cdot 1 \cdot (- 6) = 5

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 6)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 1 + 24

Add the numbers:

1 + 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 5}{2 \cdot 1}, u_{2} = \frac{- 1 - 5}{2 \cdot 1}

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

\frac{- 1 + 5}{2 \cdot 1}

Add/Subtract the numbers:

- 1 + 5 = 4

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u = \frac{- 1 - 5}{2 \cdot 1} : - 3

\frac{- 1 - 5}{2 \cdot 1}

Subtract the numbers:

- 1 - 5 = - 6

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = 2, u = - 3

Substitute back

u = tan (x)

tan (x) = 2, tan (x) = - 3

tan (x) = 2, tan (x) = - 3

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

tan (x) = - 3 : x = arctan (- 3) + πn

tan (x) = - 3

Apply trig inverse properties

tan (x) = - 3

General solutions for

tan (x) = - 3

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

x = arctan (- 3) + πn

x = arctan (- 3) + πn

Combine all the solutions

x = arctan (2) + πn, x = arctan (- 3) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = - 1.24904 \dots + πn

Popular Examples

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sin(x)=tan(x)

sin (x) = tan (x)

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+tan(x)-6=0 ?
The general solution for tan^2(x)+tan(x)-6=0 is x=1.10714…+pin,x=-1.24904…+pin