Finite Math Examples

$10 x^{2} - 36 y^{2} = 100$

Step 1

Add $36 y^{2}$ to both sides of the equation.

$10 x^{2} = 100 + 36 y^{2}$

Step 2

Divide each term in $10 x^{2} = 100 + 36 y^{2}$ by $10$ and simplify.

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Step 2.1

Divide each term in $10 x^{2} = 100 + 36 y^{2}$ by $10$ .

$\frac{10 x^{2}}{10} = \frac{100}{10} + \frac{36 y^{2}}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $10$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{10 x^{2}}{10} = \frac{100}{10} + \frac{36 y^{2}}{10}$

Step 2.2.1.2

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

$x^{2} = \frac{100}{10} + \frac{36 y^{2}}{10}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $100$ by $10$ .

$x^{2} = 10 + \frac{36 y^{2}}{10}$

Step 2.3.1.2

Cancel the common factor of $36$ and $10$ .

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Step 2.3.1.2.1

Factor $2$ out of $36 y^{2}$ .

$x^{2} = 10 + \frac{2 (18 y^{2})}{10}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $2$ out of $10$ .

$x^{2} = 10 + \frac{2 (18 y^{2})}{2 (5)}$

Step 2.3.1.2.2.2

Cancel the common factor.

$x^{2} = 10 + \frac{2 (18 y^{2})}{2 \cdot 5}$

Step 2.3.1.2.2.3

Rewrite the expression.

$x^{2} = 10 + \frac{18 y^{2}}{5}$

Step 3

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

$x = \pm \sqrt{10 + \frac{18 y^{2}}{5}}$

Step 4

Simplify $\pm \sqrt{10 + \frac{18 y^{2}}{5}}$ .

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Step 4.1

Factor $2$ out of $10 + \frac{18 y^{2}}{5}$ .

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Step 4.1.1

Factor $2$ out of $10$ .

$x = \pm \sqrt{2 (5) + \frac{18 y^{2}}{5}}$

Step 4.1.2

Factor $2$ out of $\frac{18 y^{2}}{5}$ .

$x = \pm \sqrt{2 (5) + 2 \frac{9 y^{2}}{5}}$

Step 4.1.3

Factor $2$ out of $2 (5) + 2 \frac{9 y^{2}}{5}$ .

$x = \pm \sqrt{2 (5 + \frac{9 y^{2}}{5})}$

Step 4.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \pm \sqrt{2 (5 \cdot \frac{5}{5} + \frac{9 y^{2}}{5})}$

Step 4.3

Combine $5$ and $\frac{5}{5}$ .

$x = \pm \sqrt{2 (\frac{5 \cdot 5}{5} + \frac{9 y^{2}}{5})}$

Step 4.4

Combine the numerators over the common denominator.

$x = \pm \sqrt{2 \frac{5 \cdot 5 + 9 y^{2}}{5}}$

Step 4.5

Multiply $5$ by $5$ .

$x = \pm \sqrt{2 \frac{25 + 9 y^{2}}{5}}$

Step 4.6

Combine $2$ and $\frac{25 + 9 y^{2}}{5}$ .

$x = \pm \sqrt{\frac{2 (25 + 9 y^{2})}{5}}$

Step 4.7

Rewrite $\sqrt{\frac{2 (25 + 9 y^{2})}{5}}$ as $\frac{\sqrt{2 (25 + 9 y^{2})}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})}}{\sqrt{5}}$

Step 4.8

Multiply $\frac{\sqrt{2 (25 + 9 y^{2})}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.9

Combine and simplify the denominator.

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Step 4.9.1

Multiply $\frac{\sqrt{2 (25 + 9 y^{2})}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.9.2

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 4.9.3

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 4.9.4

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.9.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.9.6

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

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Step 4.9.6.1

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.9.6.2

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.9.6.3

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

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.9.6.4

Cancel the common factor of $2$ .

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Step 4.9.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.9.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{5^{1}}$

Step 4.9.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2})} \sqrt{5}}{5}$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 (25 + 9 y^{2}) \cdot 5}}{5}$

Step 4.10.2

Multiply $5$ by $2$ .

$x = \pm \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

Step 5

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

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Step 5.1

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

$x = \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

Step 5.2

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

$x = - \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

Step 5.3

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

$x = \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

$x = - \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

$x = \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$

$x = - \frac{\sqrt{10 (25 + 9 y^{2})}}{5}$