Precalculus Examples

$x + 8 = 2 {(y - 5)}^{2}$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $2 {(y - 5)}^{2} = x + 8$ .

$2 {(y - 5)}^{2} = x + 8$

Step 1.2

Divide each term in $2 {(y - 5)}^{2} = x + 8$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 {(y - 5)}^{2} = x + 8$ by $2$ .

$\frac{2 {(y - 5)}^{2}}{2} = \frac{x}{2} + \frac{8}{2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 {(y - 5)}^{2}}{2} = \frac{x}{2} + \frac{8}{2}$

Step 1.2.2.1.2

Divide ${(y - 5)}^{2}$ by $1$ .

${(y - 5)}^{2} = \frac{x}{2} + \frac{8}{2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $8$ by $2$ .

${(y - 5)}^{2} = \frac{x}{2} + 4$

Step 1.3

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

$y - 5 = \pm \sqrt{\frac{x}{2} + 4}$

Step 1.4

Simplify $\pm \sqrt{\frac{x}{2} + 4}$ .

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Step 1.4.1

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

$y - 5 = \pm \sqrt{\frac{x}{2} + 4 \cdot \frac{2}{2}}$

Step 1.4.2

Combine $4$ and $\frac{2}{2}$ .

$y - 5 = \pm \sqrt{\frac{x}{2} + \frac{4 \cdot 2}{2}}$

Step 1.4.3

Combine the numerators over the common denominator.

$y - 5 = \pm \sqrt{\frac{x + 4 \cdot 2}{2}}$

Step 1.4.4

Multiply $4$ by $2$ .

$y - 5 = \pm \sqrt{\frac{x + 8}{2}}$

Step 1.4.5

Rewrite $\sqrt{\frac{x + 8}{2}}$ as $\frac{\sqrt{x + 8}}{\sqrt{2}}$ .

$y - 5 = \pm \frac{\sqrt{x + 8}}{\sqrt{2}}$

Step 1.4.6

Multiply $\frac{\sqrt{x + 8}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y - 5 = \pm \frac{\sqrt{x + 8}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4.7

Combine and simplify the denominator.

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Step 1.4.7.1

Multiply $\frac{\sqrt{x + 8}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.7.2

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.4.7.3

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.4.7.4

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.4.7.5

Add $1$ and $1$ .

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.4.7.6

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

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Step 1.4.7.6.1

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.4.7.6.2

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.4.7.6.3

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

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.7.6.4

Cancel the common factor of $2$ .

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Step 1.4.7.6.4.1

Cancel the common factor.

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.7.6.4.2

Rewrite the expression.

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2^{1}}$

Step 1.4.7.6.5

Evaluate the exponent.

$y - 5 = \pm \frac{\sqrt{x + 8} \sqrt{2}}{2}$

Step 1.4.8

Combine using the product rule for radicals.

$y - 5 = \pm \frac{\sqrt{(x + 8) \cdot 2}}{2}$

Step 1.4.9

Reorder factors in $\pm \frac{\sqrt{(x + 8) \cdot 2}}{2}$ .

$y - 5 = \pm \frac{\sqrt{2 (x + 8)}}{2}$

Step 1.5

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

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Step 1.5.1

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

$y - 5 = \frac{\sqrt{2 (x + 8)}}{2}$

Step 1.5.2

Add $5$ to both sides of the equation.

$y = \frac{\sqrt{2 (x + 8)}}{2} + 5$

Step 1.5.3

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

$y - 5 = - \frac{\sqrt{2 (x + 8)}}{2}$

Step 1.5.4

Add $5$ to both sides of the equation.

$y = - \frac{\sqrt{2 (x + 8)}}{2} + 5$

Step 1.5.5

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

$y = \frac{\sqrt{2 (x + 8)}}{2} + 5$

$y = - \frac{\sqrt{2 (x + 8)}}{2} + 5$

$y = \frac{\sqrt{2 (x + 8)}}{2} + 5$

$y = - \frac{\sqrt{2 (x + 8)}}{2} + 5$

$y = \frac{\sqrt{2 (x + 8)}}{2} + 5$

$y = - \frac{\sqrt{2 (x + 8)}}{2} + 5$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Reorder terms.

$y = \frac{1}{2} \cdot \sqrt{2 (x + 8)} + 5$

$y = - (\frac{1}{2} \cdot \sqrt{2 (x + 8)}) + 5$

Step 4

Remove parentheses.

$y = \frac{1}{2} \cdot \sqrt{2 (x + 8)} + 5$

$y = - \frac{1}{2} \cdot \sqrt{2 (x + 8)} + 5$

Step 5