Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $2$ to both sides of the equation.

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

Step 1.2.3

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

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

Step 1.2.4

Multiply both sides of the equation by $- 5$ .

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

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

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Step 1.2.5.1.1.1

Cancel the common factor of $5$ .

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Step 1.2.5.1.1.1.1

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

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

Step 1.2.5.1.1.1.2

Factor $5$ out of $- 5$ .

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

Step 1.2.5.1.1.1.3

Cancel the common factor.

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

Step 1.2.5.1.1.1.4

Rewrite the expression.

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

Step 1.2.5.1.1.2

Multiply.

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Step 1.2.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.1.1.2.2

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

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

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $- 5$ by $2$ .

${(x - 5)}^{2} = - 10$

Step 1.2.6

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

$x - 5 = \pm \sqrt{- 10}$

Step 1.2.7

Simplify $\pm \sqrt{- 10}$ .

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Step 1.2.7.1

Rewrite $- 10$ as $- 1 (10)$ .

$x - 5 = \pm \sqrt{- 1 (10)}$

Step 1.2.7.2

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$x - 5 = \pm \sqrt{- 1} \cdot \sqrt{10}$

Step 1.2.7.3

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

$x - 5 = \pm i \sqrt{10}$

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.

$x - 5 = i \sqrt{10}$

Step 1.2.8.2

Add $5$ to both sides of the equation.

$x = i \sqrt{10} + 5$

Step 1.2.8.3

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

$x - 5 = - i \sqrt{10}$

Step 1.2.8.4

Add $5$ to both sides of the equation.

$x = - i \sqrt{10} + 5$

Step 1.2.8.5

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

$x = i \sqrt{10} + 5, - i \sqrt{10} + 5$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

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

Step 2.2.3.1.2

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

$y = - \frac{1}{5} \cdot 25 - 2$

Step 2.2.3.1.3

Cancel the common factor of $5$ .

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Step 2.2.3.1.3.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$y = \frac{- 1}{5} \cdot 25 - 2$

Step 2.2.3.1.3.2

Factor $5$ out of $25$ .

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

Step 2.2.3.1.3.3

Cancel the common factor.

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

Step 2.2.3.1.3.4

Rewrite the expression.

$y = - 1 \cdot 5 - 2$

Step 2.2.3.1.4

Multiply $- 1$ by $5$ .

$y = - 5 - 2$

Step 2.2.3.2

Subtract $2$ from $- 5$ .

$y = - 7$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 7)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 7)$

Step 4