Find the x and y Intercepts f(x)=x^4-5x^2

$f (x) = x^{4} - 5 x^{2}$

Step 1

Find the x-intercepts.

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

$0 = x^{4} - 5 x^{2}$

Solve the equation.

Rewrite the equation as $x^{4} - 5 x^{2} = 0$ .

$x^{4} - 5 x^{2} = 0$

Factor the left side of the equation.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 5 u = 0$

Factor $u$ out of $u^{2} - 5 u$ .

Factor $u$ out of $u^{2}$ .

$u \cdot u - 5 u = 0$

Factor $u$ out of $- 5 u$ .

$u \cdot u + u \cdot - 5 = 0$

Factor $u$ out of $u \cdot u + u \cdot - 5$ .

$u (u - 5) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

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

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 5 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

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

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

$x = \sqrt{5}$

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

$x = - \sqrt{5}$

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

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

The final solution is all the values that make $x^{2} (x^{2} - 5) = 0$ true.

$x = 0, \sqrt{5}, - \sqrt{5}$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = {(0)}^{4} - 5 {(0)}^{2}$

Solve the equation.

Remove parentheses.

$y = 0^{4} - 5 {(0)}^{2}$

Remove parentheses.

$y = 0^{4} - 5 \cdot 0^{2}$

Remove parentheses.

$y = {(0)}^{4} - 5 {(0)}^{2}$

Simplify ${(0)}^{4} - 5 {(0)}^{2}$ .

Simplify each term.

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

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

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

$y = 0 - 5 \cdot 0$

Multiply $- 5$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

Find the x and y Intercepts f(x)=x^4-5x^2

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