Finite Math Examples

$\frac{{(x - 5)}^{2}}{4} + \frac{{(y + 2)}^{2}}{9} = 1$

Step 1

Subtract $\frac{{(y + 2)}^{2}}{9}$ from both sides of the equation.

$\frac{{(x - 5)}^{2}}{4} = 1 - \frac{{(y + 2)}^{2}}{9}$

Step 2

Simplify $1 - \frac{{(y + 2)}^{2}}{9}$ .

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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.

$\frac{{(x - 5)}^{2}}{4} = \frac{9}{9} - \frac{{(y + 2)}^{2}}{9}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(x - 5)}^{2}}{4} = \frac{9 - {(y + 2)}^{2}}{9}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\frac{{(x - 5)}^{2}}{4} = \frac{3^{2} - {(y + 2)}^{2}}{9}$

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 $a = 3$ and $b = y + 2$ .

$\frac{{(x - 5)}^{2}}{4} = \frac{(3 + y + 2) (3 - (y + 2))}{9}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Add $3$ and $2$ .

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (3 - (y + 2))}{9}$

Step 2.2.3.2

Apply the distributive property.

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (3 - y - 1 \cdot 2)}{9}$

Step 2.2.3.3

Multiply $- 1$ by $2$ .

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (3 - y - 2)}{9}$

Step 2.2.3.4

Subtract $2$ from $3$ .

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (- y + 1)}{9}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

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

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (- (y) + 1)}{9}$

Step 2.3.2

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

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (- (y) - 1 (- 1))}{9}$

Step 2.3.3

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

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (- (y - 1))}{9}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

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

$\frac{{(x - 5)}^{2}}{4} = \frac{(y + 5) (- 1 (y - 1))}{9}$

Step 2.3.4.2

Move the negative in front of the fraction.

$\frac{{(x - 5)}^{2}}{4} = - \frac{(y + 5) (y - 1)}{9}$

Step 3

Multiply both sides of the equation by $4$ .

$4 \frac{{(x - 5)}^{2}}{4} = 4 (- \frac{(y + 5) (y - 1)}{9})$

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 $4$ .

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Step 4.1.1.1

Cancel the common factor.

$4 \frac{{(x - 5)}^{2}}{4} = 4 (- \frac{(y + 5) (y - 1)}{9})$

Step 4.1.1.2

Rewrite the expression.

${(x - 5)}^{2} = 4 (- \frac{(y + 5) (y - 1)}{9})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $4 (- \frac{(y + 5) (y - 1)}{9})$ .

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Step 4.2.1.1

Multiply $4 (- \frac{(y + 5) (y - 1)}{9})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $4$ .

${(x - 5)}^{2} = - 4 \frac{(y + 5) (y - 1)}{9}$

Step 4.2.1.1.2

Combine $- 4$ and $\frac{(y + 5) (y - 1)}{9}$ .

${(x - 5)}^{2} = \frac{- 4 ((y + 5) (y - 1))}{9}$

Step 4.2.1.2

Move the negative in front of the fraction.

${(x - 5)}^{2} = - \frac{4 (y + 5) (y - 1)}{9}$

Step 5

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

$x - 5 = \pm \sqrt{- \frac{4 (y + 5) (y - 1)}{9}}$

Step 6

Simplify $\pm \sqrt{- \frac{4 (y + 5) (y - 1)}{9}}$ .

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Step 6.1

Rewrite $- \frac{4 (y + 5) (y - 1)}{9}$ as ${(\frac{2}{3})}^{2} \cdot (- (y + 5) (y - 1))$ .

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Step 6.1.1

Factor the perfect power $2^{2}$ out of $4 (y + 5) (y - 1)$ .

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

Step 6.1.2

Factor the perfect power $3^{2}$ out of $9$ .

$x - 5 = \pm \sqrt{- \frac{2^{2} ((y + 5) (y - 1))}{3^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{2^{2} ((y + 5) (y - 1))}{3^{2} \cdot 1}$ .

$x - 5 = \pm \sqrt{- ({(\frac{2}{3})}^{2} ((y + 5) (y - 1)))}$

Step 6.1.4

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

$x - 5 = \pm \sqrt{{(\frac{2}{3})}^{2} \cdot - 1 (y + 5) (y - 1)}$

Step 6.1.5

Add parentheses.

$x - 5 = \pm \sqrt{{(\frac{2}{3})}^{2} \cdot - 1 ((y + 5) (y - 1))}$

Step 6.1.6

Add parentheses.

$x - 5 = \pm \sqrt{{(\frac{2}{3})}^{2} \cdot (- (y + 5) (y - 1))}$

Step 6.2

Pull terms out from under the radical.

$x - 5 = \pm \frac{2}{3} \sqrt{- (y + 5) (y - 1)}$

Step 6.3

Combine $\frac{2}{3}$ and $\sqrt{- (y + 5) (y - 1)}$ .

$x - 5 = \pm \frac{2 \sqrt{- (y + 5) (y - 1)}}{3}$

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.

$x - 5 = \frac{2 \sqrt{- (y + 5) (y - 1)}}{3}$

Step 7.2

Add $5$ to both sides of the equation.

$x = \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$

Step 7.3

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

$x - 5 = - \frac{2 \sqrt{- (y + 5) (y - 1)}}{3}$

Step 7.4

Add $5$ to both sides of the equation.

$x = - \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$

Step 7.5

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

$x = \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$

$x = - \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$

$x = \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$

$x = - \frac{2 \sqrt{- (y + 5) (y - 1)}}{3} + 5$