Calculus Examples

$f (x) = 4 x^{4} - 24 x^{2}$

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 $x$ .

$0 = 4 x^{4} - 24 x^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 x^{4} - 24 x^{2} = 0$ .

$4 x^{4} - 24 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

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

$4 {(x^{2})}^{2} - 24 x^{2} = 0$

Step 1.2.2.2

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

$4 u^{2} - 24 u = 0$

Step 1.2.2.3

Factor $4 u$ out of $4 u^{2} - 24 u$ .

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Step 1.2.2.3.1

Factor $4 u$ out of $4 u^{2}$ .

$4 u (u) - 24 u = 0$

Step 1.2.2.3.2

Factor $4 u$ out of $- 24 u$ .

$4 u (u) + 4 u (- 6) = 0$

Step 1.2.2.3.3

Factor $4 u$ out of $4 u (u) + 4 u (- 6)$ .

$4 u (u - 6) = 0$

Step 1.2.2.4

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

$4 x^{2} (x^{2} - 6) = 0$

Step 1.2.3

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

$x^{2} = 0$

$x^{2} - 6 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x^{2} = 0$

Step 1.2.4.2

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

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Step 1.2.4.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.2.2.1

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

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

Step 1.2.4.2.2.2

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

$x = \pm 0$

Step 1.2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.5

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 1.2.5.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 1.2.5.2.2

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

$x = \pm \sqrt{6}$

Step 1.2.5.2.3

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

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Step 1.2.5.2.3.1

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

$x = \sqrt{6}$

Step 1.2.5.2.3.2

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

$x = - \sqrt{6}$

Step 1.2.5.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 1.2.6

The final solution is all the values that make $4 x^{2} (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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 $y$ .

$y = 4 {(0)}^{4} - 24 {(0)}^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{4} - 24 {(0)}^{2}$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot 0^{4} - 24 \cdot 0^{2}$

Step 2.2.3

Remove parentheses.

$y = 4 {(0)}^{4} - 24 {(0)}^{2}$

Step 2.2.4

Simplify $4 {(0)}^{4} - 24 {(0)}^{2}$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 4 \cdot 0 - 24 {(0)}^{2}$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 24 {(0)}^{2}$

Step 2.2.4.1.3

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

$y = 0 - 24 \cdot 0$

Step 2.2.4.1.4

Multiply $- 24$ by $0$ .

$y = 0 + 0$

Step 2.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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

Step 4