What is the general solution for 2cos^2(x)+cos(x)-10=0 ?

The general solution for 2cos^2(x)+cos(x)-10=0 is No Solution for x\in\mathbb{R}

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Popular >

2cos^2(x)+cos(x)-10=0

Solution

2 cos^{2} (x) + cos (x) - 10 = 0

Solution

No Solution for x \in R

Solution steps

2 cos^{2} (x) + cos (x) - 10 = 0

Solve by substitution

2 cos^{2} (x) + cos (x) - 10 = 0

Let:

cos (x) = u

2 u^{2} + u - 10 = 0

2 u^{2} + u - 10 = 0 : u = 2, u = - \frac{5}{2}

2 u^{2} + u - 10 = 0

Solve with the quadratic formula

2 u^{2} + u - 10 = 0

Quadratic Equation Formula:

For

a = 2, b = 1, c = - 10

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 10 )}{2 \cdot 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 10 )}{2 \cdot 2}

1^{2} - 4 \cdot 2 (- 10) = 9

1^{2} - 4 \cdot 2 (- 10)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 10)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 10

Multiply the numbers:

4 \cdot 2 \cdot 10 = 80

= 1 + 80

Add the numbers:

1 + 80 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

n a^{n} = a

9^{2} = 9

= 9

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

Separate the solutions

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

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

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

Add/Subtract the numbers:

- 1 + 9 = 8

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

u = \frac{- 1 - 9}{2 \cdot 2} : - \frac{5}{2}

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

Subtract the numbers:

- 1 - 9 = - 10

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 10}{4}

Apply the fraction rule:

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

= - \frac{10}{4}

Cancel the common factor:

2

= - \frac{5}{2}

The solutions to the quadratic equation are:

u = 2, u = - \frac{5}{2}

Substitute back

u = cos (x)

cos (x) = 2, cos (x) = - \frac{5}{2}

cos (x) = 2, cos (x) = - \frac{5}{2}

cos (x) = 2 :

No Solution

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = - \frac{5}{2} :

No Solution

cos (x) = - \frac{5}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(x)+cos(x)-10=0 ?
The general solution for 2cos^2(x)+cos(x)-10=0 is No Solution for x\in\mathbb{R}