Precalculus Examples

$y = \frac{1}{2} \cdot {(x + 5)}^{2} - 3$

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 $x$ .

$0 = \frac{1}{2} \cdot {(x + 5)}^{2} - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{2} \cdot {(x + 5)}^{2} - 3 = 0$ .

$\frac{1}{2} \cdot {(x + 5)}^{2} - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$\frac{1}{2} \cdot {(x + 5)}^{2} = 3$

Step 1.2.3

Combine $\frac{1}{2}$ and ${(x + 5)}^{2}$ .

$\frac{{(x + 5)}^{2}}{2} = 3$

Step 1.2.4

Multiply both sides of the equation by $2$ .

$2 \frac{{(x + 5)}^{2}}{2} = 2 \cdot 3$

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 $2$ .

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Step 1.2.5.1.1.1

Cancel the common factor.

$2 \frac{{(x + 5)}^{2}}{2} = 2 \cdot 3$

Step 1.2.5.1.1.2

Rewrite the expression.

${(x + 5)}^{2} = 2 \cdot 3$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $2$ by $3$ .

${(x + 5)}^{2} = 6$

Step 1.2.6

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

$x + 5 = \pm \sqrt{6}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x + 5 = \sqrt{6}$

Step 1.2.7.2

Subtract $5$ from both sides of the equation.

$x = \sqrt{6} - 5$

Step 1.2.7.3

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

$x + 5 = - \sqrt{6}$

Step 1.2.7.4

Subtract $5$ from both sides of the equation.

$x = - \sqrt{6} - 5$

Step 1.2.7.5

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

$x = \sqrt{6} - 5, - \sqrt{6} - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{6} - 5, 0), (- \sqrt{6} - 5, 0)$

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 $y$ .

$y = \frac{1}{2} \cdot {((0) + 5)}^{2} - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot {(0 + 5)}^{2} - 3$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{2} \cdot {((0) + 5)}^{2} - 3$

Step 2.2.3

Simplify $\frac{1}{2} \cdot {((0) + 5)}^{2} - 3$ .

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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 $5$ .

$y = \frac{1}{2} \cdot 5^{2} - 3$

Step 2.2.3.1.2

Raise $5$ to the power of $2$ .

$y = \frac{1}{2} \cdot 25 - 3$

Step 2.2.3.1.3

Combine $\frac{1}{2}$ and $25$ .

$y = \frac{25}{2} - 3$

Step 2.2.3.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{25}{2} - 3 \cdot \frac{2}{2}$

Step 2.2.3.3

Combine $- 3$ and $\frac{2}{2}$ .

$y = \frac{25}{2} + \frac{- 3 \cdot 2}{2}$

Step 2.2.3.4

Combine the numerators over the common denominator.

$y = \frac{25 - 3 \cdot 2}{2}$

Step 2.2.3.5

Simplify the numerator.

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Step 2.2.3.5.1

Multiply $- 3$ by $2$ .

$y = \frac{25 - 6}{2}$

Step 2.2.3.5.2

Subtract $6$ from $25$ .

$y = \frac{19}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{19}{2})$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{6} - 5, 0), (- \sqrt{6} - 5, 0)$

y-intercept(s): $(0, \frac{19}{2})$

Step 4