Finite Math Examples

$16 x^{2} + 9 y^{2} = 144$

Step 1

Subtract $9 y^{2}$ from both sides of the equation.

$16 x^{2} = 144 - 9 y^{2}$

Step 2

Divide each term in $16 x^{2} = 144 - 9 y^{2}$ by $16$ and simplify.

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Step 2.1

Divide each term in $16 x^{2} = 144 - 9 y^{2}$ by $16$ .

$\frac{16 x^{2}}{16} = \frac{144}{16} + \frac{- 9 y^{2}}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $16$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 x^{2}}{16} = \frac{144}{16} + \frac{- 9 y^{2}}{16}$

Step 2.2.1.2

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

$x^{2} = \frac{144}{16} + \frac{- 9 y^{2}}{16}$

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 $144$ by $16$ .

$x^{2} = 9 + \frac{- 9 y^{2}}{16}$

Step 2.3.1.2

Move the negative in front of the fraction.

$x^{2} = 9 - \frac{9 y^{2}}{16}$

Step 3

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

$x = \pm \sqrt{9 - \frac{9 y^{2}}{16}}$

Step 4

Simplify $\pm \sqrt{9 - \frac{9 y^{2}}{16}}$ .

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Step 4.1

Write the expression using exponents.

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Step 4.1.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2} - \frac{9 y^{2}}{16}}$

Step 4.1.2

Rewrite $\frac{9 y^{2}}{16}$ as ${(\frac{3 y}{4})}^{2}$ .

$x = \pm \sqrt{3^{2} - {(\frac{3 y}{4})}^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = \frac{3 y}{4}$ .

$x = \pm \sqrt{(3 + \frac{3 y}{4}) (3 - \frac{3 y}{4})}$

Step 4.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{(3 \cdot \frac{4}{4} + \frac{3 y}{4}) (3 - \frac{3 y}{4})}$

Step 4.4

Combine $3$ and $\frac{4}{4}$ .

$x = \pm \sqrt{(\frac{3 \cdot 4}{4} + \frac{3 y}{4}) (3 - \frac{3 y}{4})}$

Step 4.5

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{3 \cdot 4 + 3 y}{4} (3 - \frac{3 y}{4})}$

Step 4.6

Factor $3$ out of $3 \cdot 4 + 3 y$ .

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Step 4.6.1

Factor $3$ out of $3 \cdot 4$ .

$x = \pm \sqrt{\frac{3 (4) + 3 y}{4} (3 - \frac{3 y}{4})}$

Step 4.6.2

Factor $3$ out of $3 y$ .

$x = \pm \sqrt{\frac{3 (4) + 3 (y)}{4} (3 - \frac{3 y}{4})}$

Step 4.6.3

Factor $3$ out of $3 (4) + 3 (y)$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} (3 - \frac{3 y}{4})}$

Step 4.7

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} (3 \cdot \frac{4}{4} - \frac{3 y}{4})}$

Step 4.8

Combine $3$ and $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} (\frac{3 \cdot 4}{4} - \frac{3 y}{4})}$

Step 4.9

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{3 (4 + y)}{4} \cdot \frac{3 \cdot 4 - 3 y}{4}}$

Step 4.10

Factor $3$ out of $3 \cdot 4 - 3 y$ .

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Step 4.10.1

Factor $3$ out of $3 \cdot 4$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} \cdot \frac{3 (4) - 3 y}{4}}$

Step 4.10.2

Factor $3$ out of $- 3 y$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} \cdot \frac{3 (4) + 3 (- y)}{4}}$

Step 4.10.3

Factor $3$ out of $3 (4) + 3 (- y)$ .

$x = \pm \sqrt{\frac{3 (4 + y)}{4} \cdot \frac{3 (4 - y)}{4}}$

Step 4.11

Multiply $\frac{3 (4 + y)}{4}$ by $\frac{3 (4 - y)}{4}$ .

$x = \pm \sqrt{\frac{3 (4 + y) (3 (4 - y))}{4 \cdot 4}}$

Step 4.12

Multiply.

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Step 4.12.1

Multiply $3$ by $3$ .

$x = \pm \sqrt{\frac{9 (4 + y) (4 - y)}{4 \cdot 4}}$

Step 4.12.2

Multiply $4$ by $4$ .

$x = \pm \sqrt{\frac{9 (4 + y) (4 - y)}{16}}$

Step 4.13

Rewrite $\frac{9 (4 + y) (4 - y)}{16}$ as ${(\frac{3}{4})}^{2} ((2^{2} + y) (4 - y))$ .

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Step 4.13.1

Factor the perfect power $3^{2}$ out of $9 (4 + y) (4 - y)$ .

$x = \pm \sqrt{\frac{3^{2} ((2^{2} + y) (4 - y))}{16}}$

Step 4.13.2

Factor the perfect power $4^{2}$ out of $16$ .

$x = \pm \sqrt{\frac{3^{2} ((2^{2} + y) (4 - y))}{4^{2} \cdot 1}}$

Step 4.13.3

Rearrange the fraction $\frac{3^{2} ((2^{2} + y) (4 - y))}{4^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{3}{4})}^{2} ((2^{2} + y) (4 - y))}$

Step 4.14

Pull terms out from under the radical.

$x = \pm \frac{3}{4} \sqrt{(2^{2} + y) (4 - y)}$

Step 4.15

Raise $2$ to the power of $2$ .

$x = \pm \frac{3}{4} \sqrt{(4 + y) (4 - y)}$

Step 4.16

Combine $\frac{3}{4}$ and $\sqrt{(4 + y) (4 - y)}$ .

$x = \pm \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

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.

$x = \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

Step 5.2

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

$x = - \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

Step 5.3

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

$x = \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

$x = - \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

$x = \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$

$x = - \frac{3 \sqrt{(4 + y) (4 - y)}}{4}$