Algebra Examples

$x^{5} + 4 x^{3} = 0$

Step 1

Factor $x^{3}$ out of $x^{5} + 4 x^{3}$ .

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Step 1.1

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} + 4 x^{3} = 0$

Step 1.2

Factor $x^{3}$ out of $4 x^{3}$ .

$x^{3} x^{2} + x^{3} \cdot 4 = 0$

Step 1.3

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} \cdot 4$ .

$x^{3} (x^{2} + 4) = 0$

Step 2

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

$x^{3} = 0$

$x^{2} + 4 = 0$

Step 3

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

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Step 3.1

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

$x^{3} = 0$

Step 3.2

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

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Step 3.2.1

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

$x = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.2.2.2

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

$x = 0$

Step 4

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 4.2

Solve $x^{2} + 4 = 0$ for $x$ .

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 4.2.2

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

$x = \pm \sqrt{- 4}$

Step 4.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 4.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 4.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 4.2.3.3

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

$x = \pm i \cdot \sqrt{4}$

Step 4.2.3.4

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.3.5

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

$x = \pm i \cdot 2$

Step 4.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 4.2.4

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

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Step 4.2.4.1

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

$x = 2 i$

Step 4.2.4.2

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

$x = - 2 i$

Step 4.2.4.3

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

$x = 2 i, - 2 i$

Step 5

The final solution is all the values that make $x^{3} (x^{2} + 4) = 0$ true.

$x = 0, 2 i, - 2 i$

Step 6