Algebra Examples

$f (x) = - \frac{1}{2} {(x - 5)}^{2} - 3$

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

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

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

Simplify $- 2 (- \frac{{(x - 5)}^{2}}{2})$ .

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Step 1.2.5.1.1.1

Cancel the common factor of $2$ .

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Step 1.2.5.1.1.1.1

Move the leading negative in $- \frac{{(x - 5)}^{2}}{2}$ into the numerator.

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

Step 1.2.5.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 1.2.5.1.1.1.3

Cancel the common factor.

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

Step 1.2.5.1.1.1.4

Rewrite the expression.

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

Step 1.2.5.1.1.2

Multiply.

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Step 1.2.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.1.1.2.2

Multiply ${(x - 5)}^{2}$ by $1$ .

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

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

Simplify $\pm \sqrt{- 6}$ .

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Step 1.2.7.1

Rewrite $- 6$ as $- 1 (6)$ .

$x - 5 = \pm \sqrt{- 1 (6)}$

Step 1.2.7.2

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x - 5 = \pm \sqrt{- 1} \cdot \sqrt{6}$

Step 1.2.7.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

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.

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

Step 1.2.8.2

Add $5$ to both sides of the equation.

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

Step 1.2.8.3

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

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

Step 1.2.8.4

Add $5$ to both sides of the equation.

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

Step 1.2.8.5

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

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

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

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

Subtract $5$ from $0$ .

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

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

Multiply $- \frac{1}{2} \cdot 25$ .

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Step 2.2.3.1.3.1

Multiply $25$ by $- 1$ .

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

Step 2.2.3.1.3.2

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

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

Step 2.2.3.1.4

Move the negative in front of the fraction.

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

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 $- 25$ .

$y = \frac{- 31}{2}$

Step 2.2.3.6

Move the negative in front of the fraction.

$y = - \frac{31}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{31}{2})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{31}{2})$

Step 4