Calculus Examples

$f (x) = \frac{x^{2} - 8}{x + 3}$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{x^{2} - 8}{x + 3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 8 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $8$ to both sides of the equation.

$x^{2} = 8$

Step 1.2.2.2

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

$x = \pm \sqrt{8}$

Step 1.2.2.3

Simplify $\pm \sqrt{8}$ .

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Step 1.2.2.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.2.3.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 1.2.2.3.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.2.3.2

Pull terms out from under the radical.

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

Step 1.2.2.4

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

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Step 1.2.2.4.1

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

$x = 2 \sqrt{2}$

Step 1.2.2.4.2

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

$x = - 2 \sqrt{2}$

Step 1.2.2.4.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2 \sqrt{2}, 0), (- 2 \sqrt{2}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{{(0)}^{2} - 8}{(0) + 3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 8}{(0) + 3}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 8}{0 + 3}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 8}{(0) + 3}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 8}{(0) + 3}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 - 8}{0 + 3}$

Step 2.2.4.1.2

Subtract $8$ from $0$ .

$y = \frac{- 8}{0 + 3}$

Step 2.2.4.2

Simplify the expression.

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Step 2.2.4.2.1

Add $0$ and $3$ .

$y = \frac{- 8}{3}$

Step 2.2.4.2.2

Move the negative in front of the fraction.

$y = - \frac{8}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{8}{3})$

Step 3

List the intersections.

x-intercept(s): $(2 \sqrt{2}, 0), (- 2 \sqrt{2}, 0)$

y-intercept(s): $(0, - \frac{8}{3})$

Step 4