Solve by Factoring x^3+1331=0

$x^{3} + 1331 = 0$

Step 1

Rewrite $1331$ as $11^{3}$ .

$x^{3} + 11^{3} = 0$

Step 2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 11$ .

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

Step 3

Simplify.

Multiply $11$ by $- 1$ .

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

Raise $11$ to the power of $2$ .

$(x + 11) (x^{2} - 11 x + 121) = 0$

Step 4

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

$x + 11 = 0$

$x^{2} - 11 x + 121 = 0$

Step 5

Set $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 6

Set $x^{2} - 11 x + 121$ equal to $0$ and solve for $x$ .

Set $x^{2} - 11 x + 121$ equal to $0$ .

$x^{2} - 11 x + 121 = 0$

Solve $x^{2} - 11 x + 121 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 11$ , and $c = 121$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (1 \cdot 121)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 121}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 121$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 121}}{2 \cdot 1}$

Multiply $- 4$ by $121$ .

$x = \frac{11 \pm \sqrt{121 - 484}}{2 \cdot 1}$

Subtract $484$ from $121$ .

$x = \frac{11 \pm \sqrt{- 363}}{2 \cdot 1}$

Rewrite $- 363$ as $- 1 (363)$ .

$x = \frac{11 \pm \sqrt{- 1 \cdot 363}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (363)}$ as $\sqrt{- 1} \cdot \sqrt{363}$ .

$x = \frac{11 \pm \sqrt{- 1} \cdot \sqrt{363}}{2 \cdot 1}$

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

$x = \frac{11 \pm i \cdot \sqrt{363}}{2 \cdot 1}$

Rewrite $363$ as $11^{2} \cdot 3$ .

Factor $121$ out of $363$ .

$x = \frac{11 \pm i \cdot \sqrt{121 (3)}}{2 \cdot 1}$

Rewrite $121$ as $11^{2}$ .

$x = \frac{11 \pm i \cdot \sqrt{11^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{11 \pm i \cdot (11 \sqrt{3})}{2 \cdot 1}$

Move $11$ to the left of $i$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 121}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 121$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 121}}{2 \cdot 1}$

Multiply $- 4$ by $121$ .

$x = \frac{11 \pm \sqrt{121 - 484}}{2 \cdot 1}$

Subtract $484$ from $121$ .

$x = \frac{11 \pm \sqrt{- 363}}{2 \cdot 1}$

Rewrite $- 363$ as $- 1 (363)$ .

$x = \frac{11 \pm \sqrt{- 1 \cdot 363}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (363)}$ as $\sqrt{- 1} \cdot \sqrt{363}$ .

$x = \frac{11 \pm \sqrt{- 1} \cdot \sqrt{363}}{2 \cdot 1}$

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

$x = \frac{11 \pm i \cdot \sqrt{363}}{2 \cdot 1}$

Rewrite $363$ as $11^{2} \cdot 3$ .

Factor $121$ out of $363$ .

$x = \frac{11 \pm i \cdot \sqrt{121 (3)}}{2 \cdot 1}$

Rewrite $121$ as $11^{2}$ .

$x = \frac{11 \pm i \cdot \sqrt{11^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{11 \pm i \cdot (11 \sqrt{3})}{2 \cdot 1}$

Move $11$ to the left of $i$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{11 + 11 i \sqrt{3}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 1 \cdot 121}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 121$ .

Multiply $- 4$ by $1$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 121}}{2 \cdot 1}$

Multiply $- 4$ by $121$ .

$x = \frac{11 \pm \sqrt{121 - 484}}{2 \cdot 1}$

Subtract $484$ from $121$ .

$x = \frac{11 \pm \sqrt{- 363}}{2 \cdot 1}$

Rewrite $- 363$ as $- 1 (363)$ .

$x = \frac{11 \pm \sqrt{- 1 \cdot 363}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (363)}$ as $\sqrt{- 1} \cdot \sqrt{363}$ .

$x = \frac{11 \pm \sqrt{- 1} \cdot \sqrt{363}}{2 \cdot 1}$

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

$x = \frac{11 \pm i \cdot \sqrt{363}}{2 \cdot 1}$

Rewrite $363$ as $11^{2} \cdot 3$ .

Factor $121$ out of $363$ .

$x = \frac{11 \pm i \cdot \sqrt{121 (3)}}{2 \cdot 1}$

Rewrite $121$ as $11^{2}$ .

$x = \frac{11 \pm i \cdot \sqrt{11^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{11 \pm i \cdot (11 \sqrt{3})}{2 \cdot 1}$

Move $11$ to the left of $i$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{11 \pm 11 i \sqrt{3}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{11 - 11 i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = \frac{11 + 11 i \sqrt{3}}{2}, \frac{11 - 11 i \sqrt{3}}{2}$

Step 7

The final solution is all the values that make $(x + 11) (x^{2} - 11 x + 121) = 0$ true.

$x = - 11, \frac{11 + 11 i \sqrt{3}}{2}, \frac{11 - 11 i \sqrt{3}}{2}$

Solve by Factoring x^3+1331=0

Related Questions

Support LCMGCF.com by disabling your adblocker.