Algebra Examples

$x^{3} + 2 x^{2} + 5 x + 10 = 0$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(x^{3} + 2 x^{2}) + 5 x + 10 = 0$

Step 1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

Step 3

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

$x + 2 = 0$

$x^{2} + 5 = 0$

Step 4

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

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Step 4.1

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

$x + 2 = 0$

Step 4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

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Step 5.1

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

$x^{2} + 5 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$x^{2} = - 5$

Step 5.2.2

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

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

Step 5.2.3

Simplify $\pm \sqrt{- 5}$ .

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Step 5.2.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$x = \pm \sqrt{- 1 (5)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 5.2.3.3

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

$x = \pm i \sqrt{5}$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = i \sqrt{5}$

Step 5.2.4.2

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

$x = - i \sqrt{5}$

Step 5.2.4.3

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

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

Step 6

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

$x = - 2, i \sqrt{5}, - i \sqrt{5}$