Algebra Examples

$f (x) = x^{2} (x^{2} + 9) (2 x + 3) (x^{2} - 6)$

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 $x$ .

$0 = x^{2} (x^{2} + 9) (2 x + 3) (x^{2} - 6)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{2} (x^{2} + 9) (2 x + 3) (x^{2} - 6) = 0$ .

$x^{2} (x^{2} + 9) (2 x + 3) (x^{2} - 6) = 0$

Step 1.2.2

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

$x^{2} = 0$

$x^{2} + 9 = 0$

$2 x + 3 = 0$

$x^{2} - 6 = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

$x^{2} = 0$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.2.2.1

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

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

Step 1.2.3.2.2.2

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

$x = \pm 0$

Step 1.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 1.2.4.2

Solve $x^{2} + 9 = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.4.2.2

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

$x = \pm \sqrt{- 9}$

Step 1.2.4.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 1.2.4.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.4.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.4.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

Step 1.2.4.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.4.2.3.5

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

$x = \pm i \cdot 3$

Step 1.2.4.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 1.2.4.2.4

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

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Step 1.2.4.2.4.1

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

$x = 3 i$

Step 1.2.4.2.4.2

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

$x = - 3 i$

Step 1.2.4.2.4.3

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

$x = 3 i, - 3 i$

Step 1.2.5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 1.2.5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 1.2.5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 1.2.6

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

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Step 1.2.6.1

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

$x^{2} - 6 = 0$

Step 1.2.6.2

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

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Step 1.2.6.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 1.2.6.2.2

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

$x = \pm \sqrt{6}$

Step 1.2.6.2.3

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

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Step 1.2.6.2.3.1

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

$x = \sqrt{6}$

Step 1.2.6.2.3.2

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

$x = - \sqrt{6}$

Step 1.2.6.2.3.3

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

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

Step 1.2.7

The final solution is all the values that make $x^{2} (x^{2} + 9) (2 x + 3) (x^{2} - 6) = 0$ true.

$x = 0, 3 i, - 3 i, - \frac{3}{2}, \sqrt{6}, - \sqrt{6}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{3}{2}, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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 $y$ .

$y = {(0)}^{2} ({(0)}^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{2} ({(0)}^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.2

Remove parentheses.

$y = 0^{2} (0^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.3

Remove parentheses.

$y = 0^{2} (0^{2} + 9) (2 (0) + 3) (0^{2} - 6)$

Step 2.2.4

Remove parentheses.

$y = {(0)}^{2} ({(0)}^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.5

Simplify ${(0)}^{2} ({(0)}^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$ .

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Step 2.2.5.1

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

$y = 0 ({(0)}^{2} + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.5.2

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

$y = 0 (0 + 9) (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.5.3

Add $0$ and $9$ .

$y = 0 \cdot 9 (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.5.4

Multiply $0$ by $9$ .

$y = 0 (2 (0) + 3) ({(0)}^{2} - 6)$

Step 2.2.5.5

Multiply $2$ by $0$ .

$y = 0 (0 + 3) ({(0)}^{2} - 6)$

Step 2.2.5.6

Add $0$ and $3$ .

$y = 0 \cdot 3 ({(0)}^{2} - 6)$

Step 2.2.5.7

Multiply $0$ by $3$ .

$y = 0 ({(0)}^{2} - 6)$

Step 2.2.5.8

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

$y = 0 (0 - 6)$

Step 2.2.5.9

Subtract $6$ from $0$ .

$y = 0 \cdot - 6$

Step 2.2.5.10

Multiply $0$ by $- 6$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- \frac{3}{2}, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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

Step 4