Pre-Algebra Examples

$x^{2} = - 22 x - 131$

Step 1

Add $22 x$ to both sides of the equation.

$x^{2} + 22 x = - 131$

Step 2

Add $131$ to both sides of the equation.

$x^{2} + 22 x + 131 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 22$ , and $c = 131$ into the quadratic formula and solve for $x$ .

$\frac{- 22 \pm \sqrt{22^{2} - 4 \cdot (1 \cdot 131)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 1 \cdot 131}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 131$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 131}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $131$ .

$x = \frac{- 22 \pm \sqrt{484 - 524}}{2 \cdot 1}$

Step 5.1.3

Subtract $524$ from $484$ .

$x = \frac{- 22 \pm \sqrt{- 40}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 40$ as $- 1 (40)$ .

$x = \frac{- 22 \pm \sqrt{- 1 \cdot 40}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$x = \frac{- 22 \pm \sqrt{- 1} \cdot \sqrt{40}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{- 22 \pm i \cdot \sqrt{40}}{2 \cdot 1}$

Step 5.1.7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 5.1.7.1

Factor $4$ out of $40$ .

$x = \frac{- 22 \pm i \cdot \sqrt{4 (10)}}{2 \cdot 1}$

Step 5.1.7.2

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

$x = \frac{- 22 \pm i \cdot \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 22 \pm i \cdot (2 \sqrt{10})}{2 \cdot 1}$

Step 5.1.9

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

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2}$

Step 5.3

Simplify $\frac{- 22 \pm 2 i \sqrt{10}}{2}$ .

$x = - 11 \pm i \sqrt{10}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 1 \cdot 131}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 131$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 131}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $131$ .

$x = \frac{- 22 \pm \sqrt{484 - 524}}{2 \cdot 1}$

Step 6.1.3

Subtract $524$ from $484$ .

$x = \frac{- 22 \pm \sqrt{- 40}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 40$ as $- 1 (40)$ .

$x = \frac{- 22 \pm \sqrt{- 1 \cdot 40}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$x = \frac{- 22 \pm \sqrt{- 1} \cdot \sqrt{40}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{- 22 \pm i \cdot \sqrt{40}}{2 \cdot 1}$

Step 6.1.7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 6.1.7.1

Factor $4$ out of $40$ .

$x = \frac{- 22 \pm i \cdot \sqrt{4 (10)}}{2 \cdot 1}$

Step 6.1.7.2

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

$x = \frac{- 22 \pm i \cdot \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 22 \pm i \cdot (2 \sqrt{10})}{2 \cdot 1}$

Step 6.1.9

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

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2}$

Step 6.3

Simplify $\frac{- 22 \pm 2 i \sqrt{10}}{2}$ .

$x = - 11 \pm i \sqrt{10}$

Step 6.4

Change the $\pm$ to $+$ .

$x = - 11 + i \sqrt{10}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $22$ to the power of $2$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 1 \cdot 131}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 131$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 22 \pm \sqrt{484 - 4 \cdot 131}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $131$ .

$x = \frac{- 22 \pm \sqrt{484 - 524}}{2 \cdot 1}$

Step 7.1.3

Subtract $524$ from $484$ .

$x = \frac{- 22 \pm \sqrt{- 40}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 40$ as $- 1 (40)$ .

$x = \frac{- 22 \pm \sqrt{- 1 \cdot 40}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$x = \frac{- 22 \pm \sqrt{- 1} \cdot \sqrt{40}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{- 22 \pm i \cdot \sqrt{40}}{2 \cdot 1}$

Step 7.1.7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 7.1.7.1

Factor $4$ out of $40$ .

$x = \frac{- 22 \pm i \cdot \sqrt{4 (10)}}{2 \cdot 1}$

Step 7.1.7.2

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

$x = \frac{- 22 \pm i \cdot \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 22 \pm i \cdot (2 \sqrt{10})}{2 \cdot 1}$

Step 7.1.9

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

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 22 \pm 2 i \sqrt{10}}{2}$

Step 7.3

Simplify $\frac{- 22 \pm 2 i \sqrt{10}}{2}$ .

$x = - 11 \pm i \sqrt{10}$

Step 7.4

Change the $\pm$ to $-$ .

$x = - 11 - i \sqrt{10}$

Step 8

The final answer is the combination of both solutions.

$x = - 11 + i \sqrt{10}, - 11 - i \sqrt{10}$