Pre-Algebra Examples

$2 x^{2} + 6 x - 12 = 0$

Step 1

Factor $2$ out of $2 x^{2} + 6 x - 12$ .

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Step 1.1

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

$2 (x^{2}) + 6 x - 12 = 0$

Step 1.2

Factor $2$ out of $6 x$ .

$2 (x^{2}) + 2 (3 x) - 12 = 0$

Step 1.3

Factor $2$ out of $- 12$ .

$2 x^{2} + 2 (3 x) + 2 \cdot - 6 = 0$

Step 1.4

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

$2 (x^{2} + 3 x) + 2 \cdot - 6 = 0$

Step 1.5

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 6$ .

$2 (x^{2} + 3 x - 6) = 0$

Step 2

Divide each term in $2 (x^{2} + 3 x - 6) = 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 (x^{2} + 3 x - 6) = 0$ by $2$ .

$\frac{2 (x^{2} + 3 x - 6)}{2} = \frac{0}{2}$

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.

$\frac{2 (x^{2} + 3 x - 6)}{2} = \frac{0}{2}$

Step 2.2.1.2

Divide $x^{2} + 3 x - 6$ by $1$ .

$x^{2} + 3 x - 6 = \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x^{2} + 3 x - 6 = 0$

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 $a = 1$ , $b = 3$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 6}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 3 \pm \sqrt{9 + 24}}{2 \cdot 1}$

Step 5.1.3

Add $9$ and $24$ .

$x = \frac{- 3 \pm \sqrt{33}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{33}}{2}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 6}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 3 \pm \sqrt{9 + 24}}{2 \cdot 1}$

Step 6.1.3

Add $9$ and $24$ .

$x = \frac{- 3 \pm \sqrt{33}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{33}}{2}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{33}}{2}$

Step 6.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{33}}{2}$

Step 6.5

Factor $- 1$ out of $\sqrt{33}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{33})}{2}$

Step 6.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{33})$ .

$x = \frac{- 1 (3 - \sqrt{33})}{2}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{33}}{2}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 6}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 3 \pm \sqrt{9 + 24}}{2 \cdot 1}$

Step 7.1.3

Add $9$ and $24$ .

$x = \frac{- 3 \pm \sqrt{33}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm \sqrt{33}}{2}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{33}}{2}$

Step 7.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{33}}{2}$

Step 7.5

Factor $- 1$ out of $- \sqrt{33}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{33})}{2}$

Step 7.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{33})$ .

$x = \frac{- 1 (3 + \sqrt{33})}{2}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{33}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{33}}{2}, - \frac{3 + \sqrt{33}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{33}}{2}, - \frac{3 + \sqrt{33}}{2}$

Decimal Form:

$x = 1.37228132 \dots, - 4.37228132 \dots$