Finite Math Examples

$\frac{{(x - 1)}^{2}}{64} - \frac{{(y + 3)}^{2}}{36} = 1$

Step 1

Subtract $\frac{{(x - 1)}^{2}}{64}$ from both sides of the equation.

$- \frac{{(y + 3)}^{2}}{36} = 1 - \frac{{(x - 1)}^{2}}{64}$

Step 2

Simplify $1 - \frac{{(x - 1)}^{2}}{64}$ .

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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.

$- \frac{{(y + 3)}^{2}}{36} = \frac{64}{64} - \frac{{(x - 1)}^{2}}{64}$

Step 2.1.2

Combine the numerators over the common denominator.

$- \frac{{(y + 3)}^{2}}{36} = \frac{64 - {(x - 1)}^{2}}{64}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite $64$ as $8^{2}$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{8^{2} - {(x - 1)}^{2}}{64}$

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 $a = 8$ and $b = x - 1$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(8 + x - 1) (8 - (x - 1))}{64}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Subtract $1$ from $8$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (8 - (x - 1))}{64}$

Step 2.2.3.2

Apply the distributive property.

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (8 - x - - 1)}{64}$

Step 2.2.3.3

Multiply $- 1$ by $- 1$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (8 - x + 1)}{64}$

Step 2.2.3.4

Add $8$ and $1$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (- x + 9)}{64}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- x$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (- (x) + 9)}{64}$

Step 2.3.2

Rewrite $9$ as $- 1 (- 9)$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (- (x) - 1 (- 9))}{64}$

Step 2.3.3

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (- (x - 9))}{64}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$- \frac{{(y + 3)}^{2}}{36} = \frac{(x + 7) (- 1 (x - 9))}{64}$

Step 2.3.4.2

Move the negative in front of the fraction.

$- \frac{{(y + 3)}^{2}}{36} = - \frac{(x + 7) (x - 9)}{64}$

Step 3

Multiply both sides of the equation by $- 36$ .

$- 36 (- \frac{{(y + 3)}^{2}}{36}) = - 36 (- \frac{(x + 7) (x - 9)}{64})$

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

Simplify $- 36 (- \frac{{(y + 3)}^{2}}{36})$ .

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Step 4.1.1.1

Cancel the common factor of $36$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{{(y + 3)}^{2}}{36}$ into the numerator.

$- 36 \frac{- {(y + 3)}^{2}}{36} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.1.1.1.2

Factor $36$ out of $- 36$ .

$36 (- 1) \frac{- {(y + 3)}^{2}}{36} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.1.1.1.3

Cancel the common factor.

$36 \cdot - 1 \frac{- {(y + 3)}^{2}}{36} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.1.1.1.4

Rewrite the expression.

$- - {(y + 3)}^{2} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 {(y + 3)}^{2} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.1.1.2.2

Multiply ${(y + 3)}^{2}$ by $1$ .

${(y + 3)}^{2} = - 36 (- \frac{(x + 7) (x - 9)}{64})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 36 (- \frac{(x + 7) (x - 9)}{64})$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Move the leading negative in $- \frac{(x + 7) (x - 9)}{64}$ into the numerator.

${(y + 3)}^{2} = - 36 \frac{- (x + 7) (x - 9)}{64}$

Step 4.2.1.1.2

Factor $4$ out of $- 36$ .

${(y + 3)}^{2} = 4 (- 9) \frac{- (x + 7) (x - 9)}{64}$

Step 4.2.1.1.3

Factor $4$ out of $64$ .

${(y + 3)}^{2} = 4 \cdot - 9 \frac{- (x + 7) (x - 9)}{4 \cdot 16}$

Step 4.2.1.1.4

Cancel the common factor.

${(y + 3)}^{2} = 4 \cdot - 9 \frac{- (x + 7) (x - 9)}{4 \cdot 16}$

Step 4.2.1.1.5

Rewrite the expression.

${(y + 3)}^{2} = - 9 \frac{- (x + 7) (x - 9)}{16}$

Step 4.2.1.2

Combine $- 9$ and $\frac{- (x + 7) (x - 9)}{16}$ .

${(y + 3)}^{2} = \frac{- 9 (- (x + 7) (x - 9))}{16}$

Step 4.2.1.3

Multiply $- 1$ by $- 9$ .

${(y + 3)}^{2} = \frac{9 (x + 7) (x - 9)}{16}$

Step 5

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

$y + 3 = \pm \sqrt{\frac{9 (x + 7) (x - 9)}{16}}$

Step 6

Simplify $\pm \sqrt{\frac{9 (x + 7) (x - 9)}{16}}$ .

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Step 6.1

Rewrite $\frac{9 (x + 7) (x - 9)}{16}$ as ${(\frac{3}{4})}^{2} ((x + 7) (x - 9))$ .

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Step 6.1.1

Factor the perfect power $3^{2}$ out of $9 (x + 7) (x - 9)$ .

$y + 3 = \pm \sqrt{\frac{3^{2} ((x + 7) (x - 9))}{16}}$

Step 6.1.2

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

$y + 3 = \pm \sqrt{\frac{3^{2} ((x + 7) (x - 9))}{4^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{3^{2} ((x + 7) (x - 9))}{4^{2} \cdot 1}$ .

$y + 3 = \pm \sqrt{{(\frac{3}{4})}^{2} ((x + 7) (x - 9))}$

Step 6.2

Pull terms out from under the radical.

$y + 3 = \pm \frac{3}{4} \sqrt{(x + 7) (x - 9)}$

Step 6.3

Combine $\frac{3}{4}$ and $\sqrt{(x + 7) (x - 9)}$ .

$y + 3 = \pm \frac{3 \sqrt{(x + 7) (x - 9)}}{4}$

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.

$y + 3 = \frac{3 \sqrt{(x + 7) (x - 9)}}{4}$

Step 7.2

Subtract $3$ from both sides of the equation.

$y = \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$

Step 7.3

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

$y + 3 = - \frac{3 \sqrt{(x + 7) (x - 9)}}{4}$

Step 7.4

Subtract $3$ from both sides of the equation.

$y = - \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$

Step 7.5

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

$y = \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$

$y = - \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$

$y = \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$

$y = - \frac{3 \sqrt{(x + 7) (x - 9)}}{4} - 3$