What is the general solution for 6tan^2(x)-tan(x)-12=0 ?

The general solution for 6tan^2(x)-tan(x)-12=0 is x=0.98279…+pin,x=-0.92729…+pin

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Popular Trigonometry >

6tan^2(x)-tan(x)-12=0

Solution

6 tan^{2} (x) - tan (x) - 12 = 0

Solution

x = 0.98279 \dots + πn, x = - 0.92729 \dots + πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = - 53.13010 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 tan^{2} (x) - tan (x) - 12 = 0

Solve by substitution

6 tan^{2} (x) - tan (x) - 12 = 0

Let:

tan (x) = u

6 u^{2} - u - 12 = 0

6 u^{2} - u - 12 = 0 : u = \frac{3}{2}, u = - \frac{4}{3}

6 u^{2} - u - 12 = 0

Solve with the quadratic formula

6 u^{2} - u - 12 = 0

Quadratic Equation Formula:

For

a = 6, b = - 1, c = - 12

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

(- 1)^{2} - 4 \cdot 6 (- 12) = 17

(- 1)^{2} - 4 \cdot 6 (- 12)

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 6 \cdot 12

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 6 \cdot 12 = 288

4 \cdot 6 \cdot 12

Multiply the numbers:

4 \cdot 6 \cdot 12 = 288

= 288

= 1 + 288

Add the numbers:

1 + 288 = 289

= 289

Factor the number:

289 = 1 7^{2}

= 1 7^{2}

Apply radical rule:

1 7^{2} = 17

= 17

u_{1, 2} = \frac{- ( - 1 ) \pm 17}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 17}{2 \cdot 6}, u_{2} = \frac{- ( - 1 ) - 17}{2 \cdot 6}

u = \frac{- ( - 1 ) + 17}{2 \cdot 6} : \frac{3}{2}

\frac{- ( - 1 ) + 17}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 + 17}{2 \cdot 6}

Add the numbers:

1 + 17 = 18

= \frac{18}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{18}{12}

Cancel the common factor:

6

= \frac{3}{2}

u = \frac{- ( - 1 ) - 17}{2 \cdot 6} : - \frac{4}{3}

\frac{- ( - 1 ) - 17}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 - 17}{2 \cdot 6}

Subtract the numbers:

1 - 17 = - 16

= \frac{- 16}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 16}{12}

Apply the fraction rule:

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

= - \frac{16}{12}

Cancel the common factor:

4

= - \frac{4}{3}

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - \frac{4}{3}

Substitute back

u = tan (x)

tan (x) = \frac{3}{2}, tan (x) = - \frac{4}{3}

tan (x) = \frac{3}{2}, tan (x) = - \frac{4}{3}

tan (x) = \frac{3}{2} : x = arctan (\frac{3}{2}) + πn

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

tan (x) = - \frac{4}{3} : x = arctan (- \frac{4}{3}) + πn

tan (x) = - \frac{4}{3}

Apply trig inverse properties

tan (x) = - \frac{4}{3}

General solutions for

tan (x) = - \frac{4}{3}

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

x = arctan (- \frac{4}{3}) + πn

x = arctan (- \frac{4}{3}) + πn

Combine all the solutions

x = arctan (\frac{3}{2}) + πn, x = arctan (- \frac{4}{3}) + πn

Show solutions in decimal form

x = 0.98279 \dots + πn, x = - 0.92729 \dots + πn

Graph

Popular Examples

16sin^2(x)=16-8cos(x)

Frequently Asked Questions (FAQ)

What is the general solution for 6tan^2(x)-tan(x)-12=0 ?
The general solution for 6tan^2(x)-tan(x)-12=0 is x=0.98279…+pin,x=-0.92729…+pin