Precalculus Examples

$\frac{x^{2}}{576} - \frac{y^{2}}{49} = 1$

Step 1

Subtract $\frac{x^{2}}{576}$ from both sides of the equation.

$- \frac{y^{2}}{49} = 1 - \frac{x^{2}}{576}$

Step 2

Multiply both sides of the equation by $- 49$ .

$- 49 (- \frac{y^{2}}{49}) = - 49 (1 - \frac{x^{2}}{576})$

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 $- 49 (- \frac{y^{2}}{49})$ .

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Step 3.1.1.1

Cancel the common factor of $49$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{y^{2}}{49}$ into the numerator.

$- 49 \frac{- y^{2}}{49} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.1.1.1.2

Factor $49$ out of $- 49$ .

$49 (- 1) \frac{- y^{2}}{49} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.1.1.1.3

Cancel the common factor.

$49 \cdot - 1 \frac{- y^{2}}{49} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.1.1.1.4

Rewrite the expression.

$- - y^{2} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.1.1.2

Multiply.

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Step 3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y^{2} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.1.1.2.2

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

$y^{2} = - 49 (1 - \frac{x^{2}}{576})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- 49 (1 - \frac{x^{2}}{576})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = - 49 \cdot 1 - 49 (- \frac{x^{2}}{576})$

Step 3.2.1.2

Multiply $- 49$ by $1$ .

$y^{2} = - 49 - 49 (- \frac{x^{2}}{576})$

Step 3.2.1.3

Multiply $- 49 (- \frac{x^{2}}{576})$ .

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Step 3.2.1.3.1

Multiply $- 1$ by $- 49$ .

$y^{2} = - 49 + 49 \frac{x^{2}}{576}$

Step 3.2.1.3.2

Combine $49$ and $\frac{x^{2}}{576}$ .

$y^{2} = - 49 + \frac{49 x^{2}}{576}$

Step 4

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

$y = \pm \sqrt{- 49 + \frac{49 x^{2}}{576}}$

Step 5

Simplify $\pm \sqrt{- 49 + \frac{49 x^{2}}{576}}$ .

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Step 5.1

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$y = \pm \sqrt{- 49 + \frac{{(7 x)}^{2}}{576}}$

Step 5.2

Rewrite $576$ as $24^{2}$ .

$y = \pm \sqrt{- 49 + \frac{{(7 x)}^{2}}{24^{2}}}$

Step 5.3

Rewrite $\frac{{(7 x)}^{2}}{24^{2}}$ as ${(\frac{7 x}{24})}^{2}$ .

$y = \pm \sqrt{- 49 + {(\frac{7 x}{24})}^{2}}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{- 7^{2} + {(\frac{7 x}{24})}^{2}}$

Step 5.4.2

Reorder $- 7^{2}$ and ${(\frac{7 x}{24})}^{2}$ .

$y = \pm \sqrt{{(\frac{7 x}{24})}^{2} - 7^{2}}$

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 $a = \frac{7 x}{24}$ and $b = 7$ .

$y = \pm \sqrt{(\frac{7 x}{24} + 7) (\frac{7 x}{24} - 7)}$

Step 5.6

Simplify.

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Step 5.6.1

Factor $7$ out of $\frac{7 x}{24} + 7$ .

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Step 5.6.1.1

Factor $7$ out of $\frac{7 x}{24}$ .

$y = \pm \sqrt{(7 \frac{x}{24} + 7) (\frac{7 x}{24} - 7)}$

Step 5.6.1.2

Factor $7$ out of $7$ .

$y = \pm \sqrt{(7 \frac{x}{24} + 7 (1)) (\frac{7 x}{24} - 7)}$

Step 5.6.1.3

Factor $7$ out of $7 \frac{x}{24} + 7 (1)$ .

$y = \pm \sqrt{7 (\frac{x}{24} + 1) (\frac{7 x}{24} - 7)}$

Step 5.6.2

Factor $7$ out of $\frac{7 x}{24} - 7$ .

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Step 5.6.2.1

Factor $7$ out of $\frac{7 x}{24}$ .

$y = \pm \sqrt{7 (\frac{x}{24} + 1) (7 \frac{x}{24} - 7)}$

Step 5.6.2.2

Factor $7$ out of $- 7$ .

$y = \pm \sqrt{7 (\frac{x}{24} + 1) (7 \frac{x}{24} + 7 (- 1))}$

Step 5.6.2.3

Factor $7$ out of $7 \frac{x}{24} + 7 (- 1)$ .

$y = \pm \sqrt{7 (\frac{x}{24} + 1) (7 (\frac{x}{24} - 1))}$

$y = \pm \sqrt{7 (\frac{x}{24} + 1) \cdot 7 (\frac{x}{24} - 1)}$

Step 5.6.3

Multiply $7$ by $7$ .

$y = \pm \sqrt{49 (\frac{x}{24} + 1) (\frac{x}{24} - 1)}$

Step 5.7

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

$y = \pm \sqrt{49 (\frac{x}{24} + \frac{24}{24}) (\frac{x}{24} - 1)}$

Step 5.8

Combine the numerators over the common denominator.

$y = \pm \sqrt{49 \frac{x + 24}{24} (\frac{x}{24} - 1)}$

Step 5.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$y = \pm \sqrt{49 \frac{x + 24}{24} (\frac{x}{24} - 1 \cdot \frac{24}{24})}$

Step 5.10

Combine $- 1$ and $\frac{24}{24}$ .

$y = \pm \sqrt{49 \frac{x + 24}{24} (\frac{x}{24} + \frac{- 1 \cdot 24}{24})}$

Step 5.11

Combine the numerators over the common denominator.

$y = \pm \sqrt{49 \frac{x + 24}{24} \frac{x - 1 \cdot 24}{24}}$

Step 5.12

Multiply $- 1$ by $24$ .

$y = \pm \sqrt{49 \frac{x + 24}{24} \frac{x - 24}{24}}$

Step 5.13

Combine exponents.

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Step 5.13.1

Combine $49$ and $\frac{x + 24}{24}$ .

$y = \pm \sqrt{\frac{49 (x + 24)}{24} \cdot \frac{x - 24}{24}}$

Step 5.13.2

Multiply $\frac{49 (x + 24)}{24}$ by $\frac{x - 24}{24}$ .

$y = \pm \sqrt{\frac{49 (x + 24) (x - 24)}{24 \cdot 24}}$

Step 5.13.3

Multiply $24$ by $24$ .

$y = \pm \sqrt{\frac{49 (x + 24) (x - 24)}{576}}$

Step 5.14

Rewrite $\frac{49 (x + 24) (x - 24)}{576}$ as ${(\frac{7}{24})}^{2} ((x + 24) (x - 24))$ .

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Step 5.14.1

Factor the perfect power $7^{2}$ out of $49 (x + 24) (x - 24)$ .

$y = \pm \sqrt{\frac{7^{2} ((x + 24) (x - 24))}{576}}$

Step 5.14.2

Factor the perfect power $24^{2}$ out of $576$ .

$y = \pm \sqrt{\frac{7^{2} ((x + 24) (x - 24))}{24^{2} \cdot 1}}$

Step 5.14.3

Rearrange the fraction $\frac{7^{2} ((x + 24) (x - 24))}{24^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{7}{24})}^{2} ((x + 24) (x - 24))}$

Step 5.15

Pull terms out from under the radical.

$y = \pm \frac{7}{24} \sqrt{(x + 24) (x - 24)}$

Step 5.16

Combine $\frac{7}{24}$ and $\sqrt{(x + 24) (x - 24)}$ .

$y = \pm \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

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.

$y = \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

Step 6.2

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

$y = - \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

Step 6.3

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

$y = \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

$y = - \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

$y = \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

$y = - \frac{7 \sqrt{(x + 24) (x - 24)}}{24}$

Step 7

Set the radicand in $\sqrt{(x + 24) (x - 24)}$ greater than or equal to $0$ to find where the expression is defined.

$(x + 24) (x - 24) \geq 0$

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 $0$ .

$x + 24 = 0$

$x - 24 = 0$

Step 8.2

Set $x + 24$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $x + 24$ equal to $0$ .

$x + 24 = 0$

Step 8.2.2

Subtract $24$ from both sides of the equation.

$x = - 24$

Step 8.3

Set $x - 24$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $x - 24$ equal to $0$ .

$x - 24 = 0$

Step 8.3.2

Add $24$ to both sides of the equation.

$x = 24$

Step 8.4

The final solution is all the values that make $(x + 24) (x - 24) \geq 0$ true.

$x = - 24, 24$

Step 8.5

Use each root to create test intervals.

$x < - 24$

$- 24 < x < 24$

$x > 24$

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 $x < - 24$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 24$ and see if this value makes the original inequality true.

$x = - 26$

Step 8.6.1.2

Replace $x$ with $- 26$ in the original inequality.

$((- 26) + 24) ((- 26) - 24) \geq 0$

Step 8.6.1.3

The left side $100$ 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 $- 24 < x < 24$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 24 < x < 24$ 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.

$((0) + 24) ((0) - 24) \geq 0$

Step 8.6.2.3

The left side $- 576$ 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 $x > 24$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 24$ and see if this value makes the original inequality true.

$x = 26$

Step 8.6.3.2

Replace $x$ with $26$ in the original inequality.

$((26) + 24) ((26) - 24) \geq 0$

Step 8.6.3.3

The left side $100$ 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.

$x < - 24$ True

$- 24 < x < 24$ False

$x > 24$ True

$x < - 24$ True

$- 24 < x < 24$ False

$x > 24$ True

Step 8.7

The solution consists of all of the true intervals.

$x \leq - 24$ or $x \geq 24$

Step 9

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

Interval Notation:

$(- \infty, - 24] \cup [24, \infty)$

Set-Builder Notation:

${x | x \leq - 24, x \geq 24}$

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: $(- \infty, - 24] \cup [24, \infty), {x | x \leq - 24, x \geq 24}$

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

Step 12