Pre-Algebra Examples

$8 x^{2} + 5 y^{2} - 28 = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $8 x^{2}$ from both sides of the equation.

$5 y^{2} - 28 = - 8 x^{2}$

Step 1.2

Add $28$ to both sides of the equation.

$5 y^{2} = - 8 x^{2} + 28$

Step 2

Divide each term in $5 y^{2} = - 8 x^{2} + 28$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 y^{2} = - 8 x^{2} + 28$ by $5$ .

$\frac{5 y^{2}}{5} = \frac{- 8 x^{2}}{5} + \frac{28}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 y^{2}}{5} = \frac{- 8 x^{2}}{5} + \frac{28}{5}$

Step 2.2.1.2

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

$y^{2} = \frac{- 8 x^{2}}{5} + \frac{28}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{8 x^{2}}{5} + \frac{28}{5}$

Step 3

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

$y = \pm \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 4

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

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Step 4.1

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

$y = \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 4.2

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

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 4.3

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

$y = \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 5

Combine the numerators over the common denominator.

$y = \sqrt{\frac{- 8 x^{2} + 28}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 6

Factor $4$ out of $- 8 x^{2} + 28$ .

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Step 6.1

Factor $4$ out of $- 8 x^{2}$ .

$y = \sqrt{\frac{4 (- 2 x^{2}) + 28}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 6.2

Factor $4$ out of $28$ .

$y = \sqrt{\frac{4 (- 2 x^{2}) + 4 (7)}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 6.3

Factor $4$ out of $4 (- 2 x^{2}) + 4 (7)$ .

$y = \sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 7

Rewrite $\sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$ as $\frac{\sqrt{4 (- 2 x^{2} + 7)}}{\sqrt{5}}$ .

$y = \frac{\sqrt{4 (- 2 x^{2} + 7)}}{\sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{\sqrt{2^{2} (- 2 x^{2} + 7)}}{\sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 8.2

Pull terms out from under the radical.

$y = \frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 9

Multiply $\frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.3

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.4

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

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.5

Add $1$ and $1$ .

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{2}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 10.6.1

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

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6.2

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

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6.3

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

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{2}{2}}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{2}{2}}}, - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6.4.2

Rewrite the expression.

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{1}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{1}}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 10.6.5

Evaluate the exponent.

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

$y = \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 11

Combine using the product rule for radicals.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{- \frac{8 x^{2}}{5} + \frac{28}{5}}$

Step 12

Combine the numerators over the common denominator.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{\frac{- 8 x^{2} + 28}{5}}$

Step 13

Factor $4$ out of $- 8 x^{2} + 28$ .

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Step 13.1

Factor $4$ out of $- 8 x^{2}$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{\frac{4 (- 2 x^{2}) + 28}{5}}$

Step 13.2

Factor $4$ out of $28$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{\frac{4 (- 2 x^{2}) + 4 (7)}{5}}$

Step 13.3

Factor $4$ out of $4 (- 2 x^{2}) + 4 (7)$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$

Step 14

Rewrite $\sqrt{\frac{4 (- 2 x^{2} + 7)}{5}}$ as $\frac{\sqrt{4 (- 2 x^{2} + 7)}}{\sqrt{5}}$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{\sqrt{4 (- 2 x^{2} + 7)}}{\sqrt{5}}$

Step 15

Simplify the numerator.

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Step 15.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{\sqrt{2^{2} (- 2 x^{2} + 7)}}{\sqrt{5}}$

Step 15.2

Pull terms out from under the radical.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$

Step 16

Multiply $\frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - (\frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 17

Combine and simplify the denominator.

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Step 17.1

Multiply $\frac{2 \sqrt{- 2 x^{2} + 7}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 17.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 17.3

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 17.4

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

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 17.5

Add $1$ and $1$ .

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 17.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 17.6.1

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

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 17.6.2

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

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 17.6.3

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

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 17.6.4

Cancel the common factor of $2$ .

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Step 17.6.4.1

Cancel the common factor.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}, - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 17.6.4.2

Rewrite the expression.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{1}}$

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5^{1}}$

Step 17.6.5

Evaluate the exponent.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{- 2 x^{2} + 7} \sqrt{5}}{5}$

Step 18

Combine using the product rule for radicals.

$y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

$y = - \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$

Step 19

The given equation $y = \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}, - \frac{2 \sqrt{5 (- 2 x^{2} + 7)}}{5}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$