What is the general solution for -4tan^2(θ)-9tan(θ)-2=0 ?

The general solution for -4tan^2(θ)-9tan(θ)-2=0 is θ=-1.10714…+pin,θ=-0.24497…+pin

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Popular >

-4tan^2(θ)-9tan(θ)-2=0

Solution

- 4 tan^{2} (θ) - 9 tan (θ) - 2 = 0

Solution

θ = - 1.10714 \dots + πn, θ = - 0.24497 \dots + πn

Degrees

θ = - 63.43494 \dots^{\circ} + 18 0^{\circ} n, θ = - 14.03624 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 4 tan^{2} (θ) - 9 tan (θ) - 2 = 0

Solve by substitution

- 4 tan^{2} (θ) - 9 tan (θ) - 2 = 0

Let:

tan (θ) = u

- 4 u^{2} - 9 u - 2 = 0

- 4 u^{2} - 9 u - 2 = 0 : u = - 2, u = - \frac{1}{4}

- 4 u^{2} - 9 u - 2 = 0

Solve with the quadratic formula

- 4 u^{2} - 9 u - 2 = 0

Quadratic Equation Formula:

For

a = - 4, b = - 9, c = - 2

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 4 ) ( - 2 )}{2 ( - 4 )}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 4 ) ( - 2 )}{2 ( - 4 )}

(- 9)^{2} - 4 (- 4) (- 2) = 7

(- 9)^{2} - 4 (- 4) (- 2)

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} - 4 \cdot 4 \cdot 2

Multiply the numbers:

4 \cdot 4 \cdot 2 = 32

= 9^{2} - 32

9^{2} = 81

= 81 - 32

Subtract the numbers:

81 - 32 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 9 ) \pm 7}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 7}{2 ( - 4 )}, u_{2} = \frac{- ( - 9 ) - 7}{2 ( - 4 )}

u = \frac{- ( - 9 ) + 7}{2 ( - 4 )} : - 2

\frac{- ( - 9 ) + 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{9 + 7}{- 2 \cdot 4}

Add the numbers:

9 + 7 = 16

= \frac{16}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{16}{- 8}

Apply the fraction rule:

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

= - \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= - 2

u = \frac{- ( - 9 ) - 7}{2 ( - 4 )} : - \frac{1}{4}

\frac{- ( - 9 ) - 7}{2 ( - 4 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{9 - 7}{- 2 \cdot 4}

Subtract the numbers:

9 - 7 = 2

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{2}{- 8}

Apply the fraction rule:

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

= - \frac{2}{8}

Cancel the common factor:

2

= - \frac{1}{4}

The solutions to the quadratic equation are:

u = - 2, u = - \frac{1}{4}

Substitute back

u = tan (θ)

tan (θ) = - 2, tan (θ) = - \frac{1}{4}

tan (θ) = - 2, tan (θ) = - \frac{1}{4}

tan (θ) = - 2 : θ = arctan (- 2) + πn

tan (θ) = - 2

Apply trig inverse properties

tan (θ) = - 2

General solutions for

tan (θ) = - 2

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

θ = arctan (- 2) + πn

θ = arctan (- 2) + πn

tan (θ) = - \frac{1}{4} : θ = arctan (- \frac{1}{4}) + πn

tan (θ) = - \frac{1}{4}

Apply trig inverse properties

tan (θ) = - \frac{1}{4}

General solutions for

tan (θ) = - \frac{1}{4}

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

θ = arctan (- \frac{1}{4}) + πn

θ = arctan (- \frac{1}{4}) + πn

Combine all the solutions

θ = arctan (- 2) + πn, θ = arctan (- \frac{1}{4}) + πn

Show solutions in decimal form

θ = - 1.10714 \dots + πn, θ = - 0.24497 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -4tan^2(θ)-9tan(θ)-2=0 ?
The general solution for -4tan^2(θ)-9tan(θ)-2=0 is θ=-1.10714…+pin,θ=-0.24497…+pin