Algebra Examples

$x = 8 y^{2} + 5$

Step 1

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

$8 y^{2} + 5 = x$

Step 2

Subtract $5$ from both sides of the equation.

$8 y^{2} = x - 5$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $8$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{\frac{x}{8} - \frac{5}{8}}$ .

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Rewrite $\frac{x - 5}{8}$ as ${(\frac{1}{2})}^{2} \frac{x - 5}{2}$ .

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Step 5.2.1

Factor the perfect power $1^{2}$ out of $x - 5$ .

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

Step 5.2.2

Factor the perfect power $2^{2}$ out of $8$ .

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

Step 5.2.3

Rearrange the fraction $\frac{1^{2} (x - 5)}{2^{2} \cdot 2}$ .

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

Step 5.3

Pull terms out from under the radical.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

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

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

Step 5.6.5

Add $1$ and $1$ .

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

Step 5.6.6

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

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Step 5.6.6.1

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

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

Step 5.6.6.2

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

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

Step 5.6.6.3

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

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

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

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

Step 5.6.6.4.2

Rewrite the expression.

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

Step 5.6.6.5

Evaluate the exponent.

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

Step 5.7

Combine using the product rule for radicals.

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

Step 5.8

Multiply $\frac{1}{2} \cdot \frac{\sqrt{(x - 5) \cdot 2}}{2}$ .

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Step 5.8.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{(x - 5) \cdot 2}}{2}$ .

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

Step 5.8.2

Multiply $2$ by $2$ .

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

Step 5.9

Reorder factors in $\pm \frac{\sqrt{(x - 5) \cdot 2}}{4}$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

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

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

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

Step 7

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{\sqrt{2 (x - 5)}}{4}, - \frac{\sqrt{2 (x - 5)}}{4}$