Precalculus Examples

$y = x^{4} - 25$

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 $x$ .

$0 = x^{4} - 25$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{4} - 25 = 0$ .

$x^{4} - 25 = 0$

Step 1.2.2

Add $25$ to both sides of the equation.

$x^{4} = 25$

Step 1.2.3

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

$x = \pm \sqrt[4]{25}$

Step 1.2.4

Simplify $\pm \sqrt[4]{25}$ .

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Step 1.2.4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt[4]{5^{2}}$

Step 1.2.4.2

Rewrite $\sqrt[4]{5^{2}}$ as $\sqrt{\sqrt{5^{2}}}$ .

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

Step 1.2.4.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

Step 1.2.5.2

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

$x = - \sqrt{5}$

Step 1.2.5.3

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

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = {(0)}^{4} - 25$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{4} - 25$

Step 2.2.2

Remove parentheses.

$y = {(0)}^{4} - 25$

Step 2.2.3

Simplify ${(0)}^{4} - 25$ .

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 25$

Step 2.2.3.2

Subtract $25$ from $0$ .

$y = - 25$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 25)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 25)$

Step 4