What is the general solution for 4tan^2(x)+12tan(x)-27=0 ?

The general solution for 4tan^2(x)+12tan(x)-27=0 is x=0.98279…+pin,x=-1.35212…+pin

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Popular Trigonometry >

4tan^2(x)+12tan(x)-27=0

Solution

4 tan^{2} (x) + 12 tan (x) - 27 = 0

Solution

x = 0.98279 \dots + πn, x = - 1.35212 \dots + πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = - 77.47119 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) + 12 tan (x) - 27 = 0

Solve by substitution

4 tan^{2} (x) + 12 tan (x) - 27 = 0

Let:

tan (x) = u

4 u^{2} + 12 u - 27 = 0

4 u^{2} + 12 u - 27 = 0 : u = \frac{3}{2}, u = - \frac{9}{2}

4 u^{2} + 12 u - 27 = 0

Solve with the quadratic formula

4 u^{2} + 12 u - 27 = 0

Quadratic Equation Formula:

For

a = 4, b = 12, c = - 27

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 \cdot 4 ( - 27 )}{2 \cdot 4}

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 \cdot 4 ( - 27 )}{2 \cdot 4}

1 2^{2} - 4 \cdot 4 (- 27) = 24

1 2^{2} - 4 \cdot 4 (- 27)

Apply rule

- (- a) = a

= 1 2^{2} + 4 \cdot 4 \cdot 27

Multiply the numbers:

4 \cdot 4 \cdot 27 = 432

= 1 2^{2} + 432

1 2^{2} = 144

= 144 + 432

Add the numbers:

144 + 432 = 576

= 576

Factor the number:

576 = 2 4^{2}

= 2 4^{2}

Apply radical rule:

2 4^{2} = 24

= 24

u_{1, 2} = \frac{- 12 \pm 24}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 12 + 24}{2 \cdot 4}, u_{2} = \frac{- 12 - 24}{2 \cdot 4}

u = \frac{- 12 + 24}{2 \cdot 4} : \frac{3}{2}

\frac{- 12 + 24}{2 \cdot 4}

Add/Subtract the numbers:

- 12 + 24 = 12

= \frac{12}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{12}{8}

Cancel the common factor:

4

= \frac{3}{2}

u = \frac{- 12 - 24}{2 \cdot 4} : - \frac{9}{2}

\frac{- 12 - 24}{2 \cdot 4}

Subtract the numbers:

- 12 - 24 = - 36

= \frac{- 36}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 36}{8}

Apply the fraction rule:

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

= - \frac{36}{8}

Cancel the common factor:

4

= - \frac{9}{2}

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - \frac{9}{2}

Substitute back

u = tan (x)

tan (x) = \frac{3}{2}, tan (x) = - \frac{9}{2}

tan (x) = \frac{3}{2}, tan (x) = - \frac{9}{2}

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{9}{2} : x = arctan (- \frac{9}{2}) + πn

tan (x) = - \frac{9}{2}

Apply trig inverse properties

tan (x) = - \frac{9}{2}

General solutions for

tan (x) = - \frac{9}{2}

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

x = arctan (- \frac{9}{2}) + πn

x = arctan (- \frac{9}{2}) + πn

Combine all the solutions

x = arctan (\frac{3}{2}) + πn, x = arctan (- \frac{9}{2}) + πn

Show solutions in decimal form

x = 0.98279 \dots + πn, x = - 1.35212 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)+12tan(x)-27=0 ?
The general solution for 4tan^2(x)+12tan(x)-27=0 is x=0.98279…+pin,x=-1.35212…+pin