Trigonometry Examples

$k (x) = \sqrt{2 x^{2} + 5}$

Step 1

Write $k (x) = \sqrt{2 x^{2} + 5}$ as an equation.

$y = \sqrt{2 x^{2} + 5}$

Step 2

Interchange the variables.

$x = \sqrt{2 y^{2} + 5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{2 y^{2} + 5} = x$ .

$\sqrt{2 y^{2} + 5} = x$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 y^{2} + 5}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y^{2} + 5}$ as ${(2 y^{2} + 5)}^{\frac{1}{2}}$ .

${({(2 y^{2} + 5)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(2 y^{2} + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(2 y^{2} + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(2 y^{2} + 5)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(2 y^{2} + 5)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(2 y^{2} + 5)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$2 y^{2} + 5 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $5$ from both sides of the equation.

$2 y^{2} = x^{2} - 5$

Step 3.4.2

Divide each term in $2 y^{2} = x^{2} - 5$ by $2$ and simplify.

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Step 3.4.2.1

Divide each term in $2 y^{2} = x^{2} - 5$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{x^{2}}{2} + \frac{- 5}{2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{x^{2}}{2} + \frac{- 5}{2}$

Step 3.4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x^{2}}{2} + \frac{- 5}{2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Move the negative in front of the fraction.

$y^{2} = \frac{x^{2}}{2} - \frac{5}{2}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{x^{2}}{2} - \frac{5}{2}}$

Step 3.4.4

Simplify $\pm \sqrt{\frac{x^{2}}{2} - \frac{5}{2}}$ .

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Step 3.4.4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} - 5}{2}}$

Step 3.4.4.2

Rewrite $\sqrt{\frac{x^{2} - 5}{2}}$ as $\frac{\sqrt{x^{2} - 5}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{2} - 5}}{\sqrt{2}}$

Step 3.4.4.3

Multiply $\frac{\sqrt{x^{2} - 5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{2} - 5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.4.4.4

Combine and simplify the denominator.

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Step 3.4.4.4.1

Multiply $\frac{\sqrt{x^{2} - 5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.4.4.4.2

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.4.4.4.3

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4.4.4.4

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.4.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.4.4.4.6

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

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Step 3.4.4.4.6.1

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.4.4.4.6.2

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.4.4.6.3

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

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.4.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.4.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.4.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{2^{1}}$

Step 3.4.4.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x^{2} - 5} \sqrt{2}}{2}$

Step 3.4.4.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x^{2} - 5) \cdot 2}}{2}$

Step 3.4.4.6

Reorder factors in $\pm \frac{\sqrt{(x^{2} - 5) \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{2 (x^{2} - 5)}}{2}$

Step 3.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{2 (x^{2} - 5)}}{2}$

Step 3.4.5.2

Next, use the negative value of the $\pm$ to find the second solution.