Pre-Algebra Examples

$2 x^{2} + 12 x + 22 = 0$

Step 1

Factor $2$ out of $2 x^{2} + 12 x + 22$ .

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Step 1.1

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

$2 (x^{2}) + 12 x + 22 = 0$

Step 1.2

Factor $2$ out of $12 x$ .

$2 (x^{2}) + 2 (6 x) + 22 = 0$

Step 1.3

Factor $2$ out of $22$ .

$2 x^{2} + 2 (6 x) + 2 \cdot 11 = 0$

Step 1.4

Factor $2$ out of $2 x^{2} + 2 (6 x)$ .

$2 (x^{2} + 6 x) + 2 \cdot 11 = 0$

Step 1.5

Factor $2$ out of $2 (x^{2} + 6 x) + 2 \cdot 11$ .

$2 (x^{2} + 6 x + 11) = 0$

Step 2

Divide each term in $2 (x^{2} + 6 x + 11) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (x^{2} + 6 x + 11) = 0$ by $2$ .

$\frac{2 (x^{2} + 6 x + 11)}{2} = \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 6 x + 11)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $x^{2} + 6 x + 11$ by $1$ .

$x^{2} + 6 x + 11 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x^{2} + 6 x + 11 = 0$

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 $a = 1$ , $b = 6$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 11}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 6 \pm \sqrt{36 - 44}}{2 \cdot 1}$

Step 5.1.3

Subtract $44$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 5.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 5.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 5.1.9

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

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2}$

Step 5.3

Simplify $\frac{- 6 \pm 2 i \sqrt{2}}{2}$ .

$x = - 3 \pm i \sqrt{2}$

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 11}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 6 \pm \sqrt{36 - 44}}{2 \cdot 1}$

Step 6.1.3

Subtract $44$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 6.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 6.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 6.1.9

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

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2}$

Step 6.3

Simplify $\frac{- 6 \pm 2 i \sqrt{2}}{2}$ .

$x = - 3 \pm i \sqrt{2}$

Step 6.4

Change the $\pm$ to $+$ .

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

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 $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 11}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{- 6 \pm \sqrt{36 - 44}}{2 \cdot 1}$

Step 7.1.3

Subtract $44$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 7.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.1.7.1

Factor $4$ out of $8$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 7.1.7.2

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{2})}{2 \cdot 1}$

Step 7.1.9

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

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i \sqrt{2}}{2}$

Step 7.3

Simplify $\frac{- 6 \pm 2 i \sqrt{2}}{2}$ .

$x = - 3 \pm i \sqrt{2}$

Step 7.4

Change the $\pm$ to $-$ .

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

Step 8

The final answer is the combination of both solutions.

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