Trigonometry Examples

$\frac{{(x + 3)}^{2}}{25} + \frac{{(y - 1)}^{2}}{64} = 1$

Step 1

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

$\frac{{(x + 3)}^{2}}{25} = 1 - \frac{{(y - 1)}^{2}}{64}$

Step 2

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

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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.

$\frac{{(x + 3)}^{2}}{25} = \frac{64}{64} - \frac{{(y - 1)}^{2}}{64}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(x + 3)}^{2}}{25} = \frac{64 - {(y - 1)}^{2}}{64}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\frac{{(x + 3)}^{2}}{25} = \frac{8^{2} - {(y - 1)}^{2}}{64}$

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 $b = y - 1$ .

$\frac{{(x + 3)}^{2}}{25} = \frac{(8 + y - 1) (8 - (y - 1))}{64}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Subtract $1$ from $8$ .

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (8 - (y - 1))}{64}$

Step 2.2.3.2

Apply the distributive property.

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (8 - y - - 1)}{64}$

Step 2.2.3.3

Multiply $- 1$ by $- 1$ .

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (8 - y + 1)}{64}$

Step 2.2.3.4

Add $8$ and $1$ .

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (- y + 9)}{64}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

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

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (- (y) + 9)}{64}$

Step 2.3.2

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

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (- (y) - 1 (- 9))}{64}$

Step 2.3.3

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

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (- (y - 9))}{64}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

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

$\frac{{(x + 3)}^{2}}{25} = \frac{(y + 7) (- 1 (y - 9))}{64}$

Step 2.3.4.2

Move the negative in front of the fraction.

$\frac{{(x + 3)}^{2}}{25} = - \frac{(y + 7) (y - 9)}{64}$

Step 3

Multiply both sides of the equation by $25$ .

$25 \frac{{(x + 3)}^{2}}{25} = 25 (- \frac{(y + 7) (y - 9)}{64})$

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

Cancel the common factor of $25$ .

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Step 4.1.1.1

Cancel the common factor.

$25 \frac{{(x + 3)}^{2}}{25} = 25 (- \frac{(y + 7) (y - 9)}{64})$

Step 4.1.1.2

Rewrite the expression.

${(x + 3)}^{2} = 25 (- \frac{(y + 7) (y - 9)}{64})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $25 (- \frac{(y + 7) (y - 9)}{64})$ .

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Step 4.2.1.1

Multiply $25 (- \frac{(y + 7) (y - 9)}{64})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $25$ .

${(x + 3)}^{2} = - 25 \frac{(y + 7) (y - 9)}{64}$

Step 4.2.1.1.2

Combine $- 25$ and $\frac{(y + 7) (y - 9)}{64}$ .

${(x + 3)}^{2} = \frac{- 25 ((y + 7) (y - 9))}{64}$

Step 4.2.1.2

Move the negative in front of the fraction.

${(x + 3)}^{2} = - \frac{25 (y + 7) (y - 9)}{64}$

Step 5

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

$x + 3 = \pm \sqrt{- \frac{25 (y + 7) (y - 9)}{64}}$

Step 6

Simplify $\pm \sqrt{- \frac{25 (y + 7) (y - 9)}{64}}$ .

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Step 6.1

Rewrite $- \frac{25 (y + 7) (y - 9)}{64}$ as ${(\frac{5}{8})}^{2} \cdot (- (y + 7) (y - 9))$ .

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Step 6.1.1

Factor the perfect power $5^{2}$ out of $25 (y + 7) (y - 9)$ .

$x + 3 = \pm \sqrt{- \frac{5^{2} ((y + 7) (y - 9))}{64}}$

Step 6.1.2

Factor the perfect power $8^{2}$ out of $64$ .

$x + 3 = \pm \sqrt{- \frac{5^{2} ((y + 7) (y - 9))}{8^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{5^{2} ((y + 7) (y - 9))}{8^{2} \cdot 1}$ .

$x + 3 = \pm \sqrt{- ({(\frac{5}{8})}^{2} ((y + 7) (y - 9)))}$

Step 6.1.4

Reorder $- 1$ and ${(\frac{5}{8})}^{2}$ .

$x + 3 = \pm \sqrt{{(\frac{5}{8})}^{2} \cdot - 1 (y + 7) (y - 9)}$

Step 6.1.5

Add parentheses.

$x + 3 = \pm \sqrt{{(\frac{5}{8})}^{2} \cdot - 1 ((y + 7) (y - 9))}$

Step 6.1.6

Add parentheses.

$x + 3 = \pm \sqrt{{(\frac{5}{8})}^{2} \cdot (- (y + 7) (y - 9))}$

Step 6.2

Pull terms out from under the radical.

$x + 3 = \pm \frac{5}{8} \sqrt{- (y + 7) (y - 9)}$

Step 6.3

Combine $\frac{5}{8}$ and $\sqrt{- (y + 7) (y - 9)}$ .

$x + 3 = \pm \frac{5 \sqrt{- (y + 7) (y - 9)}}{8}$

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.

$x + 3 = \frac{5 \sqrt{- (y + 7) (y - 9)}}{8}$

Step 7.2

Subtract $3$ from both sides of the equation.

$x = \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$

Step 7.3

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

$x + 3 = - \frac{5 \sqrt{- (y + 7) (y - 9)}}{8}$

Step 7.4

Subtract $3$ from both sides of the equation.

$x = - \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$

Step 7.5

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

$x = \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$

$x = - \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$

$x = \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$

$x = - \frac{5 \sqrt{- (y + 7) (y - 9)}}{8} - 3$