Precalculus Examples

$2 x^{2} + 24 x + 2 y^{2} = 0$

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 $x$ .

$2 x^{2} + 24 x + 2 {(0)}^{2} = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $2 x^{2} + 24 x + 2 {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$2 x^{2} + 24 x + 2 \cdot 0 = 0$

Step 1.2.1.1.2

Multiply $2$ by $0$ .

$2 x^{2} + 24 x + 0 = 0$

Step 1.2.1.2

Add $2 x^{2} + 24 x$ and $0$ .

$2 x^{2} + 24 x = 0$

Step 1.2.2

Factor $2 x$ out of $2 x^{2} + 24 x$ .

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Step 1.2.2.1

Factor $2 x$ out of $2 x^{2}$ .

$2 x (x) + 24 x = 0$

Step 1.2.2.2

Factor $2 x$ out of $24 x$ .

$2 x (x) + 2 x (12) = 0$

Step 1.2.2.3

Factor $2 x$ out of $2 x (x) + 2 x (12)$ .

$2 x (x + 12) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 12 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x + 12$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Step 1.2.5.2

Subtract $12$ from both sides of the equation.

$x = - 12$

Step 1.2.6

The final solution is all the values that make $2 x (x + 12) = 0$ true.

$x = 0, - 12$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 12, 0)$

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 $y$ .

$2 {(0)}^{2} + 24 (0) + 2 y^{2} = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $2 {(0)}^{2} + 24 (0) + 2 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$2 \cdot 0 + 24 (0) + 2 y^{2} = 0$

Step 2.2.1.1.2

Multiply $2$ by $0$ .

$0 + 24 (0) + 2 y^{2} = 0$

Step 2.2.1.1.3

Multiply $24$ by $0$ .

$0 + 0 + 2 y^{2} = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 0 + 2 y^{2}$ .

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + 2 y^{2} = 0$

Step 2.2.1.2.2

Add $0$ and $2 y^{2}$ .

$2 y^{2} = 0$

Step 2.2.2

Divide each term in $2 y^{2} = 0$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 y^{2} = 0$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{0}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{0}{2}$

Step 2.2.2.2.1.2

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

$y^{2} = \frac{0}{2}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $2$ .

$y^{2} = 0$

Step 2.2.3

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

$y = \pm \sqrt{0}$

Step 2.2.4

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.1

Rewrite $0$ as $0^{2}$ .

$y = \pm \sqrt{0^{2}}$

Step 2.2.4.2

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

$y = \pm 0$

Step 2.2.4.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 12, 0)$

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

Step 4