Algebra Examples

$f (x) = 2 (x^{2} + 1) (x^{2} - 3)$

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 $x$ .

$0 = 2 (x^{2} + 1) (x^{2} - 3)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 (x^{2} + 1) (x^{2} - 3) = 0$ .

$2 (x^{2} + 1) (x^{2} - 3) = 0$

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 $0$ .

$x^{2} + 1 = 0$

$x^{2} - 3 = 0$

Step 1.2.3

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 1.2.3.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.3.2.2

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

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

Step 1.2.3.2.3

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

$x = \pm i$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

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

$x = i$

Step 1.2.3.2.4.2

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

$x = - i$

Step 1.2.3.2.4.3

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

$x = i, - i$

Step 1.2.4

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 1.2.4.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 1.2.4.2.2

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

$x = \pm \sqrt{3}$

Step 1.2.4.2.3

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

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Step 1.2.4.2.3.1

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

$x = \sqrt{3}$

Step 1.2.4.2.3.2

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

$x = - \sqrt{3}$

Step 1.2.4.2.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 1.2.5

The final solution is all the values that make $2 (x^{2} + 1) (x^{2} - 3) = 0$ true.

$x = i, - i, \sqrt{3}, - \sqrt{3}$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = 2 ({(0)}^{2} + 1) ({(0)}^{2} - 3)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 (0^{2} + 1) ({(0)}^{2} - 3)$

Step 2.2.2

Remove parentheses.

$y = 2 (0^{2} + 1) (0^{2} - 3)$

Step 2.2.3

Remove parentheses.

$y = 2 ({(0)}^{2} + 1) ({(0)}^{2} - 3)$

Step 2.2.4

Simplify $2 ({(0)}^{2} + 1) ({(0)}^{2} - 3)$ .

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Step 2.2.4.1

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

$y = 2 (0 + 1) ({(0)}^{2} - 3)$

Step 2.2.4.2

Add $0$ and $1$ .

$y = 2 \cdot 1 ({(0)}^{2} - 3)$

Step 2.2.4.3

Multiply $2$ by $1$ .

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

Step 2.2.4.4

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

$y = 2 (0 - 3)$

Step 2.2.4.5

Subtract $3$ from $0$ .

$y = 2 \cdot - 3$

Step 2.2.4.6

Multiply $2$ by $- 3$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 6)$

Step 4