Algebra Examples

$f (x) = \frac{1}{4} {(x + 2)}^{2} - 9$

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 $x$ .

$0 = \frac{1}{4} \cdot {(x + 2)}^{2} - 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{4} \cdot {(x + 2)}^{2} - 9 = 0$ .

$\frac{1}{4} \cdot {(x + 2)}^{2} - 9 = 0$

Step 1.2.2

Add $9$ to both sides of the equation.

$\frac{1}{4} \cdot {(x + 2)}^{2} = 9$

Step 1.2.3

Combine $\frac{1}{4}$ and ${(x + 2)}^{2}$ .

$\frac{{(x + 2)}^{2}}{4} = 9$

Step 1.2.4

Multiply both sides of the equation by $4$ .

$4 \frac{{(x + 2)}^{2}}{4} = 4 \cdot 9$

Step 1.2.5

Simplify both sides of the equation.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Cancel the common factor of $4$ .

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Step 1.2.5.1.1.1

Cancel the common factor.

$4 \frac{{(x + 2)}^{2}}{4} = 4 \cdot 9$

Step 1.2.5.1.1.2

Rewrite the expression.

${(x + 2)}^{2} = 4 \cdot 9$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $4$ by $9$ .

${(x + 2)}^{2} = 36$

Step 1.2.6

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

$x + 2 = \pm \sqrt{36}$

Step 1.2.7

Simplify $\pm \sqrt{36}$ .

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Step 1.2.7.1

Rewrite $36$ as $6^{2}$ .

$x + 2 = \pm \sqrt{6^{2}}$

Step 1.2.7.2

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

$x + 2 = \pm 6$

Step 1.2.8

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

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Step 1.2.8.1

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

$x + 2 = 6$

Step 1.2.8.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.8.2.1

Subtract $2$ from both sides of the equation.

$x = 6 - 2$

Step 1.2.8.2.2

Subtract $2$ from $6$ .

$x = 4$

Step 1.2.8.3

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

$x + 2 = - 6$

Step 1.2.8.4

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.8.4.1

Subtract $2$ from both sides of the equation.

$x = - 6 - 2$

Step 1.2.8.4.2

Subtract $2$ from $- 6$ .

$x = - 8$

Step 1.2.8.5

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

$x = 4, - 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0), (- 8, 0)$

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 $y$ .

$y = \frac{1}{4} \cdot {((0) + 2)}^{2} - 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{4} \cdot {(0 + 2)}^{2} - 9$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{4} \cdot {((0) + 2)}^{2} - 9$

Step 2.2.3

Simplify $\frac{1}{4} \cdot {((0) + 2)}^{2} - 9$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $2$ .

$y = \frac{1}{4} \cdot 2^{2} - 9$

Step 2.2.3.1.2

Raise $2$ to the power of $2$ .

$y = \frac{1}{4} \cdot 4 - 9$

Step 2.2.3.1.3

Cancel the common factor of $4$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

$y = \frac{1}{4} \cdot 4 - 9$

Step 2.2.3.1.3.2

Rewrite the expression.

$y = 1 - 9$

Step 2.2.3.2

Subtract $9$ from $1$ .

$y = - 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 8)$

Step 3

List the intersections.

x-intercept(s): $(4, 0), (- 8, 0)$

y-intercept(s): $(0, - 8)$

Step 4