Precalculus Examples

$y = 2 x^{2} - 18$

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 $x$ .

$0 = 2 x^{2} - 18$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{2} - 18 = 0$ .

$2 x^{2} - 18 = 0$

Step 1.2.2

Add $18$ to both sides of the equation.

$2 x^{2} = 18$

Step 1.2.3

Divide each term in $2 x^{2} = 18$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x^{2} = 18$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{18}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{18}{2}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{18}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $18$ by $2$ .

$x^{2} = 9$

Step 1.2.4

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

$x = \pm \sqrt{9}$

Step 1.2.5

Simplify $\pm \sqrt{9}$ .

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Step 1.2.5.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 1.2.5.2

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

$x = \pm 3$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = 3$

Step 1.2.6.2

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

$x = - 3$

Step 1.2.6.3

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

$x = 3, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 3, 0)$

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 $y$ .

$y = 2 {(0)}^{2} - 18$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} - 18$

Step 2.2.2

Remove parentheses.

$y = 2 {(0)}^{2} - 18$

Step 2.2.3

Simplify $2 {(0)}^{2} - 18$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = 2 \cdot 0 - 18$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 - 18$

Step 2.2.3.2

Subtract $18$ from $0$ .

$y = - 18$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 18)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 3, 0)$

y-intercept(s): $(0, - 18)$

Step 4