What is the general solution for cos^2(θ)+9cos(θ)+18=0 ?

The general solution for cos^2(θ)+9cos(θ)+18=0 is No Solution for θ\in\mathbb{R}

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Popular >

cos^2(θ)+9cos(θ)+18=0

Solution

cos^{2} (θ) + 9 cos (θ) + 18 = 0

Solution

No Solution for θ \in R

Solution steps

cos^{2} (θ) + 9 cos (θ) + 18 = 0

Solve by substitution

cos^{2} (θ) + 9 cos (θ) + 18 = 0

Let:

cos (θ) = u

u^{2} + 9 u + 18 = 0

u^{2} + 9 u + 18 = 0 : u = - 3, u = - 6

u^{2} + 9 u + 18 = 0

Solve with the quadratic formula

u^{2} + 9 u + 18 = 0

Quadratic Equation Formula:

For

a = 1, b = 9, c = 18

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 1 \cdot 18}{2 \cdot 1}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 1 \cdot 18}{2 \cdot 1}

9^{2} - 4 \cdot 1 \cdot 18 = 3

9^{2} - 4 \cdot 1 \cdot 18

Multiply the numbers:

4 \cdot 1 \cdot 18 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

Subtract the numbers:

81 - 72 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 9 \pm 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 9 + 3}{2 \cdot 1}, u_{2} = \frac{- 9 - 3}{2 \cdot 1}

u = \frac{- 9 + 3}{2 \cdot 1} : - 3

\frac{- 9 + 3}{2 \cdot 1}

Add/Subtract the numbers:

- 9 + 3 = - 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

u = \frac{- 9 - 3}{2 \cdot 1} : - 6

\frac{- 9 - 3}{2 \cdot 1}

Subtract the numbers:

- 9 - 3 = - 12

= \frac{- 12}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12}{2}

Apply the fraction rule:

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

= - \frac{12}{2}

Divide the numbers:

\frac{12}{2} = 6

= - 6

The solutions to the quadratic equation are:

u = - 3, u = - 6

Substitute back

u = cos (θ)

cos (θ) = - 3, cos (θ) = - 6

cos (θ) = - 3, cos (θ) = - 6

cos (θ) = - 3 :

No Solution

cos (θ) = - 3

- 1 \leq cos (x) \leq 1

No Solution

cos (θ) = - 6 :

No Solution

cos (θ) = - 6

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(θ)+9cos(θ)+18=0 ?
The general solution for cos^2(θ)+9cos(θ)+18=0 is No Solution for θ\in\mathbb{R}