Algebra Examples

$f (x) = \frac{- 2 x^{2} + 32}{3 x - 12}$

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 $x$ .

$0 = \frac{- 2 x^{2} + 32}{3 x - 12}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$- 2 x^{2} + 32 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $32$ from both sides of the equation.

$- 2 x^{2} = - 32$

Step 1.2.2.2

Divide each term in $- 2 x^{2} = - 32$ by $- 2$ and simplify.

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Step 1.2.2.2.1

Divide each term in $- 2 x^{2} = - 32$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 32}{- 2}$

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 $- 2$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 32}{- 2}$

Step 1.2.2.2.2.1.2

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

$x^{2} = \frac{- 32}{- 2}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $- 32$ by $- 2$ .

$x^{2} = 16$

Step 1.2.2.3

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

$x = \pm \sqrt{16}$

Step 1.2.2.4

Simplify $\pm \sqrt{16}$ .

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Step 1.2.2.4.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 1.2.2.4.2

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

$x = \pm 4$

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.

$x = 4$

Step 1.2.2.5.2

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

$x = - 4$

Step 1.2.2.5.3

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

$x = 4, - 4$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{- 2 x^{2} + 32}{3 x - 12}$ true.

$x = - 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 4, 0)$

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 $y$ .

$y = \frac{- 2 {(0)}^{2} + 32}{3 (0) - 12}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{- 2 \cdot 0^{2} + 32}{3 (0) - 12}$

Step 2.2.2

Remove parentheses.

$y = \frac{- 2 {(0)}^{2} + 32}{3 (0) - 12}$

Step 2.2.3

Simplify $\frac{- 2 {(0)}^{2} + 32}{3 (0) - 12}$ .

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Step 2.2.3.1

Cancel the common factor of $- 2 {(0)}^{2} + 32$ and $3 (0) - 12$ .

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Step 2.2.3.1.1

Factor $2$ out of $- 2 {(0)}^{2}$ .

$y = \frac{2 (- {(0)}^{2}) + 32}{3 (0) - 12}$

Step 2.2.3.1.2

Factor $2$ out of $32$ .

$y = \frac{2 (- {(0)}^{2}) + 2 (16)}{3 (0) - 12}$

Step 2.2.3.1.3

Factor $2$ out of $2 (- {(0)}^{2}) + 2 (16)$ .

$y = \frac{2 (- {(0)}^{2} + 16)}{3 (0) - 12}$

Step 2.2.3.1.4

Cancel the common factors.

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Step 2.2.3.1.4.1

Factor $2$ out of $3 (0)$ .

$y = \frac{2 (- {(0)}^{2} + 16)}{2 (3 \cdot (0)) - 12}$

Step 2.2.3.1.4.2

Factor $2$ out of $- 12$ .

$y = \frac{2 (- {(0)}^{2} + 16)}{2 (3 \cdot (0)) + 2 (- 6)}$

Step 2.2.3.1.4.3

Factor $2$ out of $2 (3 \cdot (0)) + 2 (- 6)$ .

$y = \frac{2 (- {(0)}^{2} + 16)}{2 (3 \cdot (0) - 6)}$

Step 2.2.3.1.4.4

Cancel the common factor.

$y = \frac{2 (- {(0)}^{2} + 16)}{2 (3 \cdot (0) - 6)}$

Step 2.2.3.1.4.5

Rewrite the expression.

$y = \frac{- {(0)}^{2} + 16}{3 \cdot (0) - 6}$

Step 2.2.3.2

Simplify the numerator.

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Step 2.2.3.2.1

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

$y = \frac{- 0 + 16}{3 \cdot (0) - 6}$

Step 2.2.3.2.2

Multiply $- 1$ by $0$ .

$y = \frac{0 + 16}{3 \cdot (0) - 6}$

Step 2.2.3.2.3

Add $0$ and $16$ .

$y = \frac{16}{3 \cdot (0) - 6}$

Step 2.2.3.3

Simplify the denominator.

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Step 2.2.3.3.1

Multiply $3$ by $0$ .

$y = \frac{16}{0 - 6}$

Step 2.2.3.3.2

Subtract $6$ from $0$ .

$y = \frac{16}{- 6}$

Step 2.2.3.4

Reduce the expression by cancelling the common factors.

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Step 2.2.3.4.1

Cancel the common factor of $16$ and $- 6$ .

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Step 2.2.3.4.1.1

Factor $2$ out of $16$ .

$y = \frac{2 (8)}{- 6}$

Step 2.2.3.4.1.2

Cancel the common factors.

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Step 2.2.3.4.1.2.1

Factor $2$ out of $- 6$ .

$y = \frac{2 \cdot 8}{2 \cdot - 3}$

Step 2.2.3.4.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 8}{2 \cdot - 3}$

Step 2.2.3.4.1.2.3

Rewrite the expression.

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

Step 2.2.3.4.2

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): $(- 4, 0)$

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

Step 4