Precalculus Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $1$ to both sides of the equation.

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

Step 1.2.3

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

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

Step 1.2.4

Any root of $1$ is $1$ .

$x - 5 = \pm 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$x - 5 = 1$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Add $5$ to both sides of the equation.

$x = 1 + 5$

Step 1.2.5.2.2

Add $1$ and $5$ .

$x = 6$

Step 1.2.5.3

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

$x - 5 = - 1$

Step 1.2.5.4

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

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Step 1.2.5.4.1

Add $5$ to both sides of the equation.

$x = - 1 + 5$

Step 1.2.5.4.2

Add $- 1$ and $5$ .

$x = 4$

Step 1.2.5.5

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

$x = 6, 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0), (4, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

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

Step 2.2.3.1.2

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

$y = 25 - 1$

Step 2.2.3.2

Subtract $1$ from $25$ .

$y = 24$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 24)$

Step 3

List the intersections.

x-intercept(s): $(6, 0), (4, 0)$

y-intercept(s): $(0, 24)$

Step 4