Algebra Examples

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

Step 1

Write ${(x - 2)}^{2} - 5$ as an equation.

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

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = {(x - 2)}^{2} - 5$

Step 2.2

Solve the equation.

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Step 2.2.1

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

${(x - 2)}^{2} - 5 = 0$

Step 2.2.2

Add $5$ to both sides of the equation.

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

Step 2.2.3

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

$x - 2 = \pm \sqrt{5}$

Step 2.2.4

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

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Step 2.2.4.1

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

$x - 2 = \sqrt{5}$

Step 2.2.4.2

Add $2$ to both sides of the equation.

$x = \sqrt{5} + 2$

Step 2.2.4.3

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

$x - 2 = - \sqrt{5}$

Step 2.2.4.4

Add $2$ to both sides of the equation.

$x = - \sqrt{5} + 2$

Step 2.2.4.5

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

$x = \sqrt{5} + 2, - \sqrt{5} + 2$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{5} + 2, 0), (- \sqrt{5} + 2, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = {((0) - 2)}^{2} - 5$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = {(0 - 2)}^{2} - 5$

Step 3.2.2

Remove parentheses.

$y = {((0) - 2)}^{2} - 5$

Step 3.2.3

Simplify ${((0) - 2)}^{2} - 5$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Subtract $2$ from $0$ .

$y = {(- 2)}^{2} - 5$

Step 3.2.3.1.2

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

$y = 4 - 5$

Step 3.2.3.2

Subtract $5$ from $4$ .

$y = - 1$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 4

List the intersections.

x-intercept(s): $(\sqrt{5} + 2, 0), (- \sqrt{5} + 2, 0)$

y-intercept(s): $(0, - 1)$

Step 5