Precalculus Examples

$r (x) = \frac{3 x^{2} + 6}{x^{2} - 2 x - 3}$

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 $x$ .

$0 = \frac{3 x^{2} + 6}{x^{2} - 2 x - 3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$3 x^{2} + 6 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $6$ from both sides of the equation.

$3 x^{2} = - 6$

Step 1.2.2.2

Divide each term in $3 x^{2} = - 6$ by $3$ and simplify.

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Step 1.2.2.2.1

Divide each term in $3 x^{2} = - 6$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 6}{3}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 6}{3}$

Step 1.2.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 6}{3}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $- 6$ by $3$ .

$x^{2} = - 2$

Step 1.2.2.3

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

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

Step 1.2.2.4

Simplify $\pm \sqrt{- 2}$ .

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Step 1.2.2.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 1.2.2.4.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 1.2.2.4.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 1.2.2.5

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

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Step 1.2.2.5.1

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

$x = i \sqrt{2}$

Step 1.2.2.5.2

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

$x = - i \sqrt{2}$

Step 1.2.2.5.3

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

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

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = \frac{3 {(0)}^{2} + 6}{{(0)}^{2} - 2 \cdot 0 - 3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{3 \cdot 0^{2} + 6}{{(0)}^{2} - 2 \cdot 0 - 3}$

Step 2.2.2

Remove parentheses.

$y = \frac{3 \cdot 0^{2} + 6}{0^{2} - 2 \cdot 0 - 3}$

Step 2.2.3

Remove parentheses.

$y = \frac{3 {(0)}^{2} + 6}{{(0)}^{2} - 2 \cdot 0 - 3}$

Step 2.2.4

Simplify $\frac{3 {(0)}^{2} + 6}{{(0)}^{2} - 2 \cdot 0 - 3}$ .

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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 $0$ .

$y = \frac{3 \cdot 0 + 6}{0^{2} - 2 \cdot 0 - 3}$

Step 2.2.4.1.2

Multiply $3$ by $0$ .

$y = \frac{0 + 6}{0^{2} - 2 \cdot 0 - 3}$

Step 2.2.4.1.3

Add $0$ and $6$ .

$y = \frac{6}{0^{2} - 2 \cdot 0 - 3}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{6}{0 - 2 \cdot 0 - 3}$

Step 2.2.4.2.2

Multiply $- 2$ by $0$ .

$y = \frac{6}{0 + 0 - 3}$

Step 2.2.4.2.3

Add $0$ and $0$ .

$y = \frac{6}{0 - 3}$

Step 2.2.4.2.4

Subtract $3$ from $0$ .

$y = \frac{6}{- 3}$

Step 2.2.4.3

Divide $6$ by $- 3$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 2)$

Step 4