Finite Math Examples

$\frac{{(x + 4)}^{2}}{9} + \frac{{(y - 2)}^{2}}{16} = 1$

Step 1

Subtract $\frac{{(y - 2)}^{2}}{16}$ from both sides of the equation.

$\frac{{(x + 4)}^{2}}{9} = 1 - \frac{{(y - 2)}^{2}}{16}$

Step 2

Simplify $1 - \frac{{(y - 2)}^{2}}{16}$ .

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Step 2.1

Combine into one fraction.

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Step 2.1.1

Write $1$ as a fraction with a common denominator.

$\frac{{(x + 4)}^{2}}{9} = \frac{16}{16} - \frac{{(y - 2)}^{2}}{16}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(x + 4)}^{2}}{9} = \frac{16 - {(y - 2)}^{2}}{16}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{4^{2} - {(y - 2)}^{2}}{16}$

Step 2.2.2

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

$\frac{{(x + 4)}^{2}}{9} = \frac{(4 + y - 2) (4 - (y - 2))}{16}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Subtract $2$ from $4$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (4 - (y - 2))}{16}$

Step 2.2.3.2

Apply the distributive property.

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (4 - y - - 2)}{16}$

Step 2.2.3.3

Multiply $- 1$ by $- 2$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (4 - y + 2)}{16}$

Step 2.2.3.4

Add $4$ and $2$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (- y + 6)}{16}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- y$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (- (y) + 6)}{16}$

Step 2.3.2

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (- (y) - 1 (- 6))}{16}$

Step 2.3.3

Factor $- 1$ out of $- (y) - 1 (- 6)$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (- (y - 6))}{16}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Rewrite $- (y - 6)$ as $- 1 (y - 6)$ .

$\frac{{(x + 4)}^{2}}{9} = \frac{(y + 2) (- 1 (y - 6))}{16}$

Step 2.3.4.2

Move the negative in front of the fraction.

$\frac{{(x + 4)}^{2}}{9} = - \frac{(y + 2) (y - 6)}{16}$

Step 3

Multiply both sides of the equation by $9$ .

$9 \frac{{(x + 4)}^{2}}{9} = 9 (- \frac{(y + 2) (y - 6)}{16})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $9$ .

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Step 4.1.1.1

Cancel the common factor.

$9 \frac{{(x + 4)}^{2}}{9} = 9 (- \frac{(y + 2) (y - 6)}{16})$

Step 4.1.1.2

Rewrite the expression.

${(x + 4)}^{2} = 9 (- \frac{(y + 2) (y - 6)}{16})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $9 (- \frac{(y + 2) (y - 6)}{16})$ .

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Step 4.2.1.1

Multiply $9 (- \frac{(y + 2) (y - 6)}{16})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $9$ .

${(x + 4)}^{2} = - 9 \frac{(y + 2) (y - 6)}{16}$

Step 4.2.1.1.2

Combine $- 9$ and $\frac{(y + 2) (y - 6)}{16}$ .

${(x + 4)}^{2} = \frac{- 9 ((y + 2) (y - 6))}{16}$

Step 4.2.1.2

Move the negative in front of the fraction.

${(x + 4)}^{2} = - \frac{9 (y + 2) (y - 6)}{16}$

Step 5

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

$x + 4 = \pm \sqrt{- \frac{9 (y + 2) (y - 6)}{16}}$

Step 6

Simplify $\pm \sqrt{- \frac{9 (y + 2) (y - 6)}{16}}$ .

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Step 6.1

Rewrite $- \frac{9 (y + 2) (y - 6)}{16}$ as ${(\frac{3}{2^{2}})}^{2} \cdot (- (y + 2) (y - 6))$ .

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Step 6.1.1

Factor the perfect power $3^{2}$ out of $9 (y + 2) (y - 6)$ .

$x + 4 = \pm \sqrt{- \frac{3^{2} ((y + 2) (y - 6))}{16}}$

Step 6.1.2

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

$x + 4 = \pm \sqrt{- \frac{3^{2} ((y + 2) (y - 6))}{4^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{3^{2} ((y + 2) (y - 6))}{4^{2} \cdot 1}$ .

$x + 4 = \pm \sqrt{- ({(\frac{3}{4})}^{2} ((y + 2) (y - 6)))}$

Step 6.1.4

Reorder $- 1$ and ${(\frac{3}{4})}^{2}$ .

$x + 4 = \pm \sqrt{{(\frac{3}{4})}^{2} \cdot - 1 (y + 2) (y - 6)}$

Step 6.1.5

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

$x + 4 = \pm \sqrt{{(\frac{3}{2^{2}})}^{2} \cdot - 1 (y + 2) (y - 6)}$

Step 6.1.6

Add parentheses.

$x + 4 = \pm \sqrt{{(\frac{3}{2^{2}})}^{2} \cdot - 1 ((y + 2) (y - 6))}$

Step 6.1.7

Add parentheses.

$x + 4 = \pm \sqrt{{(\frac{3}{2^{2}})}^{2} \cdot (- (y + 2) (y - 6))}$

Step 6.2

Pull terms out from under the radical.

$x + 4 = \pm \frac{3}{2^{2}} \sqrt{- (y + 2) (y - 6)}$

Step 6.3

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

$x + 4 = \pm \frac{3}{4} \sqrt{- (y + 2) (y - 6)}$

Step 6.4

Combine $\frac{3}{4}$ and $\sqrt{- (y + 2) (y - 6)}$ .

$x + 4 = \pm \frac{3 \sqrt{- (y + 2) (y - 6)}}{4}$

Step 7

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

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Step 7.1

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

$x + 4 = \frac{3 \sqrt{- (y + 2) (y - 6)}}{4}$

Step 7.2

Subtract $4$ from both sides of the equation.

$x = \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$

Step 7.3

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

$x + 4 = - \frac{3 \sqrt{- (y + 2) (y - 6)}}{4}$

Step 7.4

Subtract $4$ from both sides of the equation.

$x = - \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$

Step 7.5

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

$x = \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$

$x = - \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$

$x = \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$

$x = - \frac{3 \sqrt{- (y + 2) (y - 6)}}{4} - 4$