What is the general solution for 3cos^2(x)+2cos(x)-8=0 ?

The general solution for 3cos^2(x)+2cos(x)-8=0 is No Solution for x\in\mathbb{R}

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Popular >

3cos^2(x)+2cos(x)-8=0

Solution

3 cos^{2} (x) + 2 cos (x) - 8 = 0

Solution

No Solution for x \in R

Solution steps

3 cos^{2} (x) + 2 cos (x) - 8 = 0

Solve by substitution

3 cos^{2} (x) + 2 cos (x) - 8 = 0

Let:

cos (x) = u

3 u^{2} + 2 u - 8 = 0

3 u^{2} + 2 u - 8 = 0 : u = \frac{4}{3}, u = - 2

3 u^{2} + 2 u - 8 = 0

Solve with the quadratic formula

3 u^{2} + 2 u - 8 = 0

Quadratic Equation Formula:

For

a = 3, b = 2, c = - 8

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 3 ( - 8 )}{2 \cdot 3}

2^{2} - 4 \cdot 3 (- 8) = 10

2^{2} - 4 \cdot 3 (- 8)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 3 \cdot 8

Multiply the numbers:

4 \cdot 3 \cdot 8 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

Add the numbers:

4 + 96 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 2 + 10}{2 \cdot 3}, u_{2} = \frac{- 2 - 10}{2 \cdot 3}

u = \frac{- 2 + 10}{2 \cdot 3} : \frac{4}{3}

\frac{- 2 + 10}{2 \cdot 3}

Add/Subtract the numbers:

- 2 + 10 = 8

= \frac{8}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

u = \frac{- 2 - 10}{2 \cdot 3} : - 2

\frac{- 2 - 10}{2 \cdot 3}

Subtract the numbers:

- 2 - 10 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = cos (x)

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

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

cos (x) = \frac{4}{3} :

No Solution

cos (x) = \frac{4}{3}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = - 2 :

No Solution

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos^2(x)+2cos(x)-8=0 ?
The general solution for 3cos^2(x)+2cos(x)-8=0 is No Solution for x\in\mathbb{R}