Precalculus Examples

$2 \cos^{2} (x) - \cos (x) - 1 = 0$

Step 1

Substitute $u$ for $\cos (x)$ .

$2 {(u)}^{2} - (u) - 1 = 0$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 2.1.1

Factor $- 1$ out of $- u$ .

$2 u^{2} - u - 1 = 0$

Step 2.1.2

Rewrite $- 1$ as $1$ plus $- 2$

$2 u^{2} + (1 - 2) u - 1 = 0$

Step 2.1.3

Apply the distributive property.

$2 u^{2} + 1 u - 2 u - 1 = 0$

Step 2.1.4

Multiply $u$ by $1$ .

$2 u^{2} + u - 2 u - 1 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$2 u^{2} + u - 2 u - 1 = 0$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$u (2 u + 1) - (2 u + 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(2 u + 1) (u - 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 u + 1 = 0$

$u - 1 = 0$

Step 4

Set $2 u + 1$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $2 u + 1$ equal to $0$ .

$2 u + 1 = 0$

Step 4.2

Solve $2 u + 1 = 0$ for $u$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Step 4.2.2

Divide each term in $2 u = - 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 u = - 1$ by $2$ .

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{2}$

Step 5

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6

The final solution is all the values that make $(2 u + 1) (u - 1) = 0$ true.

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

Step 7

Substitute $\cos (x)$ for $u$ .

$\cos (x) = - \frac{1}{2}, 1$

Step 8

Set up each of the solutions to solve for $x$ .

$\cos (x) = - \frac{1}{2}$

$\cos (x) = 1$

Step 9

Solve for $x$ in $\cos (x) = - \frac{1}{2}$ .

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Step 9.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{1}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 9.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{2 π}{3}$

Step 9.4

Simplify $2 π - \frac{2 π}{3}$ .

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Step 9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 9.4.2

Combine fractions.

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Step 9.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 9.4.3

Simplify the numerator.

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Step 9.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 9.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 9.5

Find the period of $\cos (x)$ .

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Step 9.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 10

Solve for $x$ in $\cos (x) = 1$ .

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Step 10.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (1)$

Step 10.2

Simplify the right side.

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Step 10.2.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 10.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - 0$

Step 10.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 10.5

Find the period of $\cos (x)$ .

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Step 10.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 10.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 10.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 11

List all of the solutions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 12

Consolidate the answers.

$x = \frac{2 π n}{3}$ , for any integer $n$