Precalculus Examples

$\frac{x^{2} - 25}{x^{2} - 16}$

Step 1

Write $\frac{x^{2} - 25}{x^{2} - 16}$ as an equation.

$y = \frac{x^{2} - 25}{x^{2} - 16}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{x^{2} - 25}{x^{2} - 16}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$x^{2} - 25 = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Add $25$ to both sides of the equation.

$x^{2} = 25$

Step 2.2.2.2

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

$x = \pm \sqrt{25}$

Step 2.2.2.3

Simplify $\pm \sqrt{25}$ .

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Step 2.2.2.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 2.2.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

Step 2.2.2.4

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

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Step 2.2.2.4.1

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

$x = 5$

Step 2.2.2.4.2

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

$x = - 5$

Step 2.2.2.4.3

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

$x = 5, - 5$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- 5, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{{(0)}^{2} - 25}{{(0)}^{2} - 16}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{0^{2} - 25}{{(0)}^{2} - 16}$

Step 3.2.2

Remove parentheses.

$y = \frac{0^{2} - 25}{0^{2} - 16}$

Step 3.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 25}{{(0)}^{2} - 16}$

Step 3.2.4

Simplify $\frac{{(0)}^{2} - 25}{{(0)}^{2} - 16}$ .

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Step 3.2.4.1

Simplify the numerator.

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Step 3.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 - 25}{0^{2} - 16}$

Step 3.2.4.1.2

Subtract $25$ from $0$ .

$y = \frac{- 25}{0^{2} - 16}$

Step 3.2.4.2

Simplify the denominator.

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Step 3.2.4.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{- 25}{0 - 16}$

Step 3.2.4.2.2

Subtract $16$ from $0$ .

$y = \frac{- 25}{- 16}$

Step 3.2.4.3

Dividing two negative values results in a positive value.

$y = \frac{25}{16}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{25}{16})$

Step 4

List the intersections.

x-intercept(s): $(5, 0), (- 5, 0)$

y-intercept(s): $(0, \frac{25}{16})$

Step 5