Algebra Examples

$- x^{3} - x^{2} = 11 x + 11$

Step 1

Subtract $11 x$ from both sides of the equation.

$- x^{3} - x^{2} - 11 x = 11$

Step 2

Subtract $11$ from both sides of the equation.

$- x^{3} - x^{2} - 11 x - 11 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 1$ out of $- x^{3} - x^{2} - 11 x - 11$ .

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Step 3.1.1

Factor $- 1$ out of $- x^{3}$ .

$- (x^{3}) - x^{2} - 11 x - 11 = 0$

Step 3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{3}) - (x^{2}) - 11 x - 11 = 0$

Step 3.1.3

Factor $- 1$ out of $- 11 x$ .

$- (x^{3}) - (x^{2}) - (11 x) - 11 = 0$

Step 3.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$- (x^{3}) - (x^{2}) - (11 x) - 1 \cdot 11 = 0$

Step 3.1.5

Factor $- 1$ out of $- (x^{3}) - (x^{2})$ .

$- (x^{3} + x^{2}) - (11 x) - 1 \cdot 11 = 0$

Step 3.1.6

Factor $- 1$ out of $- (x^{3} + x^{2}) - (11 x)$ .

$- (x^{3} + x^{2} + 11 x) - 1 \cdot 11 = 0$

Step 3.1.7

Factor $- 1$ out of $- (x^{3} + x^{2} + 11 x) - 1 (11)$ .

$- (x^{3} + x^{2} + 11 x + 11) = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$- ((x^{3} + x^{2}) + 11 x + 11) = 0$

Step 3.2.2

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

$- (x^{2} (x + 1) + 11 (x + 1)) = 0$

Step 3.3

Factor.

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Step 3.3.1

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$- ((x + 1) (x^{2} + 11)) = 0$

Step 3.3.2

Remove unnecessary parentheses.

$- (x + 1) (x^{2} + 11) = 0$

Step 4

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

$x + 1 = 0$

$x^{2} + 11 = 0$

Step 5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

Set $x^{2} + 11$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} + 11$ equal to $0$ .

$x^{2} + 11 = 0$

Step 6.2

Solve $x^{2} + 11 = 0$ for $x$ .

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Step 6.2.1

Subtract $11$ from both sides of the equation.

$x^{2} = - 11$

Step 6.2.2

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

$x = \pm \sqrt{- 11}$

Step 6.2.3

Simplify $\pm \sqrt{- 11}$ .

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Step 6.2.3.1

Rewrite $- 11$ as $- 1 (11)$ .

$x = \pm \sqrt{- 1 (11)}$

Step 6.2.3.2

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{11}$

Step 6.2.3.3

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

$x = \pm i \sqrt{11}$

Step 6.2.4

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

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Step 6.2.4.1

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

$x = i \sqrt{11}$

Step 6.2.4.2

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

$x = - i \sqrt{11}$

Step 6.2.4.3

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

$x = i \sqrt{11}, - i \sqrt{11}$

Step 7

The final solution is all the values that make $- (x + 1) (x^{2} + 11) = 0$ true.

$x = - 1, i \sqrt{11}, - i \sqrt{11}$

Step 8