Algebra Examples

$\sqrt{2} \sin^{2} (x) + \cos (x) = 0$

Step 1

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $\sqrt{2}$ by $1$ .

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

Step 3

Reorder the polynomial.

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

Step 4

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

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

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - \sqrt{2}$ , $b = 1$ , and $c = \sqrt{2}$ into the quadratic formula and solve for $u$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- \sqrt{2} \cdot \sqrt{2})}}{2 (- \sqrt{2})}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$u = \frac{- 1 \pm \sqrt{1 - 4 \cdot (- 1 \sqrt{2}) \cdot \sqrt{2}}}{2 (- \sqrt{2})}$

Step 7.1.2

Multiply $- 4$ by $- 1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 \sqrt{2} \cdot \sqrt{2}}}{2 (- \sqrt{2})}$

Step 7.1.3

Multiply $4 \sqrt{2} \sqrt{2}$ .

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Step 7.1.3.1

Raise $\sqrt{2}$ to the power of $1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 (\sqrt{2} \sqrt{2})}}{2 (- \sqrt{2})}$

Step 7.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 (\sqrt{2} \sqrt{2})}}{2 (- \sqrt{2})}$

Step 7.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{- 1 \pm \sqrt{1 + 4 {\sqrt{2}}^{1 + 1}}}{2 (- \sqrt{2})}$

Step 7.1.3.4

Add $1$ and $1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 {\sqrt{2}}^{2}}}{2 (- \sqrt{2})}$

Step 7.1.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 {(2^{\frac{1}{2}})}^{2}}}{2 (- \sqrt{2})}$

Step 7.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 2^{\frac{1}{2} \cdot 2}}}{2 (- \sqrt{2})}$

Step 7.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 2^{\frac{2}{2}}}}{2 (- \sqrt{2})}$

Step 7.1.4.4

Cancel the common factor of $2$ .

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Step 7.1.4.4.1

Cancel the common factor.

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 2^{\frac{2}{2}}}}{2 (- \sqrt{2})}$

Step 7.1.4.4.2

Rewrite the expression.

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 2}}{2 (- \sqrt{2})}$

Step 7.1.4.5

Evaluate the exponent.

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 2}}{2 (- \sqrt{2})}$

Step 7.1.5

Multiply $4$ by $2$ .

$u = \frac{- 1 \pm \sqrt{1 + 8}}{2 (- \sqrt{2})}$

Step 7.1.6

Add $1$ and $8$ .

$u = \frac{- 1 \pm \sqrt{9}}{2 (- \sqrt{2})}$

Step 7.1.7

Rewrite $9$ as $3^{2}$ .

$u = \frac{- 1 \pm \sqrt{3^{2}}}{2 (- \sqrt{2})}$

Step 7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 1 \pm 3}{2 (- \sqrt{2})}$

Step 7.2

Multiply $- 1$ by $2$ .

$u = \frac{- 1 \pm 3}{- 2 \sqrt{2}}$

Step 7.3

Simplify $\frac{- 1 \pm 3}{- 2 \sqrt{2}}$ .

$u = \frac{1 \pm 3}{2 \sqrt{2}}$

Step 7.4

Multiply $\frac{1 \pm 3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$u = \frac{1 \pm 3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.5

Combine and simplify the denominator.

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Step 7.5.1

Multiply $\frac{1 \pm 3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 7.5.2

Move $\sqrt{2}$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 7.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 7.5.4

Raise $\sqrt{2}$ to the power of $1$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 7.5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{(1 \pm 3) \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 7.5.6

Add $1$ and $1$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 7.5.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 7.5.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 7.5.7.3

Combine $\frac{1}{2}$ and $2$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 7.5.7.4

Cancel the common factor of $2$ .

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Step 7.5.7.4.1

Cancel the common factor.

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 7.5.7.4.2

Rewrite the expression.

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \cdot 2}$

Step 7.5.7.5

Evaluate the exponent.

$u = \frac{(1 \pm 3) \sqrt{2}}{2 \cdot 2}$

Step 7.6

Multiply $2$ by $2$ .

$u = \frac{(1 \pm 3) \sqrt{2}}{4}$

Step 7.7

Reorder factors in $\frac{(1 \pm 3) \sqrt{2}}{4}$ .

$u = \frac{\sqrt{2} (1 \pm 3)}{4}$

Step 8

The final answer is the combination of both solutions.

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

Step 9

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

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

Step 10

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

$\cos (x) = \sqrt{2}$

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

Step 11

Solve for $x$ in $\cos (x) = \sqrt{2}$ .

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Step 11.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $\sqrt{2}$ does not fall in this range, there is no solution.

No solution

Step 12

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

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Step 12.1

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

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

Step 12.2

Simplify the right side.

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Step 12.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 12.3

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.

$x = 2 π - \frac{3 π}{4}$

Step 12.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 12.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$