Precalculus Examples

$\frac{x^{2}}{225} - \frac{y^{2}}{64} = 1$

Step 1

Subtract $\frac{x^{2}}{225}$ from both sides of the equation.

$- \frac{y^{2}}{64} = 1 - \frac{x^{2}}{225}$

Step 2

Multiply both sides of the equation by $- 64$ .

$- 64 (- \frac{y^{2}}{64}) = - 64 (1 - \frac{x^{2}}{225})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor of $64$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{y^{2}}{64}$ into the numerator.

$- 64 \frac{- y^{2}}{64} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.1.1.1.2

Factor $64$ out of $- 64$ .

$64 (- 1) \frac{- y^{2}}{64} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.1.1.1.3

Cancel the common factor.

$64 \cdot - 1 \frac{- y^{2}}{64} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.1.1.1.4

Rewrite the expression.

$- - y^{2} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.1.1.2

Multiply.

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Step 3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y^{2} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.1.1.2.2

Multiply $y^{2}$ by $1$ .

$y^{2} = - 64 (1 - \frac{x^{2}}{225})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- 64 (1 - \frac{x^{2}}{225})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = - 64 \cdot 1 - 64 (- \frac{x^{2}}{225})$

Step 3.2.1.2

Multiply $- 64$ by $1$ .

$y^{2} = - 64 - 64 (- \frac{x^{2}}{225})$

Step 3.2.1.3

Multiply $- 64 (- \frac{x^{2}}{225})$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 64$ .

$y^{2} = - 64 + 64 \frac{x^{2}}{225}$

Step 3.2.1.3.2

Combine $64$ and $\frac{x^{2}}{225}$ .

$y^{2} = - 64 + \frac{64 x^{2}}{225}$

Step 4

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

$y = \pm \sqrt{- 64 + \frac{64 x^{2}}{225}}$

Step 5

Simplify $\pm \sqrt{- 64 + \frac{64 x^{2}}{225}}$ .

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Step 5.1

Rewrite $64 x^{2}$ as ${(8 x)}^{2}$ .

$y = \pm \sqrt{- 64 + \frac{{(8 x)}^{2}}{225}}$

Step 5.2

Rewrite $225$ as $15^{2}$ .

$y = \pm \sqrt{- 64 + \frac{{(8 x)}^{2}}{15^{2}}}$

Step 5.3

Rewrite $\frac{{(8 x)}^{2}}{15^{2}}$ as ${(\frac{8 x}{15})}^{2}$ .

$y = \pm \sqrt{- 64 + {(\frac{8 x}{15})}^{2}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

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

$y = \pm \sqrt{- 8^{2} + {(\frac{8 x}{15})}^{2}}$

Step 5.4.2

Reorder $- 8^{2}$ and ${(\frac{8 x}{15})}^{2}$ .

$y = \pm \sqrt{{(\frac{8 x}{15})}^{2} - 8^{2}}$

Step 5.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{8 x}{15}$ and $b = 8$ .

$y = \pm \sqrt{(\frac{8 x}{15} + 8) (\frac{8 x}{15} - 8)}$

Step 5.6

Simplify.

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Step 5.6.1

Factor $8$ out of $\frac{8 x}{15} + 8$ .

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Step 5.6.1.1

Factor $8$ out of $\frac{8 x}{15}$ .

$y = \pm \sqrt{(8 \frac{x}{15} + 8) (\frac{8 x}{15} - 8)}$

Step 5.6.1.2

Factor $8$ out of $8$ .

$y = \pm \sqrt{(8 \frac{x}{15} + 8 (1)) (\frac{8 x}{15} - 8)}$

Step 5.6.1.3

Factor $8$ out of $8 \frac{x}{15} + 8 (1)$ .

$y = \pm \sqrt{8 (\frac{x}{15} + 1) (\frac{8 x}{15} - 8)}$

Step 5.6.2

Factor $8$ out of $\frac{8 x}{15} - 8$ .

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Step 5.6.2.1

Factor $8$ out of $\frac{8 x}{15}$ .

$y = \pm \sqrt{8 (\frac{x}{15} + 1) (8 \frac{x}{15} - 8)}$

Step 5.6.2.2

Factor $8$ out of $- 8$ .

$y = \pm \sqrt{8 (\frac{x}{15} + 1) (8 \frac{x}{15} + 8 (- 1))}$

Step 5.6.2.3

Factor $8$ out of $8 \frac{x}{15} + 8 (- 1)$ .

$y = \pm \sqrt{8 (\frac{x}{15} + 1) (8 (\frac{x}{15} - 1))}$

$y = \pm \sqrt{8 (\frac{x}{15} + 1) \cdot 8 (\frac{x}{15} - 1)}$

Step 5.6.3

Multiply $8$ by $8$ .

$y = \pm \sqrt{64 (\frac{x}{15} + 1) (\frac{x}{15} - 1)}$

Step 5.7

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

$y = \pm \sqrt{64 (\frac{x}{15} + \frac{15}{15}) (\frac{x}{15} - 1)}$

Step 5.8

Combine the numerators over the common denominator.

$y = \pm \sqrt{64 \frac{x + 15}{15} (\frac{x}{15} - 1)}$

Step 5.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$y = \pm \sqrt{64 \frac{x + 15}{15} (\frac{x}{15} - 1 \cdot \frac{15}{15})}$

Step 5.10

Combine $- 1$ and $\frac{15}{15}$ .

$y = \pm \sqrt{64 \frac{x + 15}{15} (\frac{x}{15} + \frac{- 1 \cdot 15}{15})}$

Step 5.11

Combine the numerators over the common denominator.

$y = \pm \sqrt{64 \frac{x + 15}{15} \frac{x - 1 \cdot 15}{15}}$

Step 5.12

Multiply $- 1$ by $15$ .

$y = \pm \sqrt{64 \frac{x + 15}{15} \frac{x - 15}{15}}$

Step 5.13

Combine exponents.

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Step 5.13.1

Combine $64$ and $\frac{x + 15}{15}$ .

$y = \pm \sqrt{\frac{64 (x + 15)}{15} \cdot \frac{x - 15}{15}}$

Step 5.13.2

Multiply $\frac{64 (x + 15)}{15}$ by $\frac{x - 15}{15}$ .

$y = \pm \sqrt{\frac{64 (x + 15) (x - 15)}{15 \cdot 15}}$

Step 5.13.3

Multiply $15$ by $15$ .

$y = \pm \sqrt{\frac{64 (x + 15) (x - 15)}{225}}$

Step 5.14

Rewrite $\frac{64 (x + 15) (x - 15)}{225}$ as ${(\frac{8}{15})}^{2} ((x + 15) (x - 15))$ .

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Step 5.14.1

Factor the perfect power $8^{2}$ out of $64 (x + 15) (x - 15)$ .

$y = \pm \sqrt{\frac{8^{2} ((x + 15) (x - 15))}{225}}$

Step 5.14.2

Factor the perfect power $15^{2}$ out of $225$ .

$y = \pm \sqrt{\frac{8^{2} ((x + 15) (x - 15))}{15^{2} \cdot 1}}$

Step 5.14.3

Rearrange the fraction $\frac{8^{2} ((x + 15) (x - 15))}{15^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{8}{15})}^{2} ((x + 15) (x - 15))}$

Step 5.15

Pull terms out from under the radical.

$y = \pm \frac{8}{15} \sqrt{(x + 15) (x - 15)}$

Step 5.16

Combine $\frac{8}{15}$ and $\sqrt{(x + 15) (x - 15)}$ .

$y = \pm \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

Step 6

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

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Step 6.1

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

$y = \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

Step 6.2

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

$y = - \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

Step 6.3

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

$y = \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

$y = - \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

$y = \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

$y = - \frac{8 \sqrt{(x + 15) (x - 15)}}{15}$

Step 7

Set the radicand in $\sqrt{(x + 15) (x - 15)}$ greater than or equal to $0$ to find where the expression is defined.

$(x + 15) (x - 15) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 15 = 0$

$x - 15 = 0$

Step 8.2

Set $x + 15$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $x + 15$ equal to $0$ .

$x + 15 = 0$

Step 8.2.2

Subtract $15$ from both sides of the equation.

$x = - 15$

Step 8.3

Set $x - 15$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $x - 15$ equal to $0$ .

$x - 15 = 0$

Step 8.3.2

Add $15$ to both sides of the equation.

$x = 15$

Step 8.4

The final solution is all the values that make $(x + 15) (x - 15) \geq 0$ true.

$x = - 15, 15$

Step 8.5

Use each root to create test intervals.

$x < - 15$

$- 15 < x < 15$

$x > 15$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < - 15$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 15$ and see if this value makes the original inequality true.

$x = - 18$

Step 8.6.1.2

Replace $x$ with $- 18$ in the original inequality.

$((- 18) + 15) ((- 18) - 15) \geq 0$

Step 8.6.1.3

The left side $99$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.2

Test a value on the interval $- 15 < x < 15$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 15 < x < 15$ and see if this value makes the original inequality true.

$x = 0$

Step 8.6.2.2

Replace $x$ with $0$ in the original inequality.

$((0) + 15) ((0) - 15) \geq 0$

Step 8.6.2.3

The left side $- 225$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.3

Test a value on the interval $x > 15$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 15$ and see if this value makes the original inequality true.

$x = 18$

Step 8.6.3.2

Replace $x$ with $18$ in the original inequality.

$((18) + 15) ((18) - 15) \geq 0$

Step 8.6.3.3

The left side $99$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 15$ True

$- 15 < x < 15$ False

$x > 15$ True

$x < - 15$ True

$- 15 < x < 15$ False

$x > 15$ True

Step 8.7

The solution consists of all of the true intervals.

$x \leq - 15$ or $x \geq 15$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 15] \cup [15, \infty)$

Set-Builder Notation:

${x | x \leq - 15, x \geq 15}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 11

Determine the domain and range.

Domain: $(- \infty, - 15] \cup [15, \infty), {x | x \leq - 15, x \geq 15}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 12