Pre-Algebra Examples

$x (x + 12) = 182$

Step 1

Simplify $x (x + 12)$ .

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Step 1.1

Apply the distributive property.

$x \cdot x + x \cdot 12 = 182$

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 12 = 182$

Step 1.2.2

Move $12$ to the left of $x$ .

$x^{2} + 12 x = 182$

Step 2

Subtract $182$ from both sides of the equation.

$x^{2} + 12 x - 182 = 0$

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 $a = 1$ , $b = 12$ , and $c = - 182$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (1 \cdot - 182)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 182}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 182$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 182}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 182$ .

$x = \frac{- 12 \pm \sqrt{144 + 728}}{2 \cdot 1}$

Step 5.1.3

Add $144$ and $728$ .

$x = \frac{- 12 \pm \sqrt{872}}{2 \cdot 1}$

Step 5.1.4

Rewrite $872$ as $2^{2} \cdot 218$ .

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Step 5.1.4.1

Factor $4$ out of $872$ .

$x = \frac{- 12 \pm \sqrt{4 (218)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 218}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{218}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 2 \sqrt{218}}{2}$

Step 5.3

Simplify $\frac{- 12 \pm 2 \sqrt{218}}{2}$ .

$x = - 6 \pm \sqrt{218}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 182}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 182$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 182}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 182$ .

$x = \frac{- 12 \pm \sqrt{144 + 728}}{2 \cdot 1}$

Step 6.1.3

Add $144$ and $728$ .

$x = \frac{- 12 \pm \sqrt{872}}{2 \cdot 1}$

Step 6.1.4

Rewrite $872$ as $2^{2} \cdot 218$ .

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Step 6.1.4.1

Factor $4$ out of $872$ .

$x = \frac{- 12 \pm \sqrt{4 (218)}}{2 \cdot 1}$

Step 6.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 218}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{218}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 2 \sqrt{218}}{2}$

Step 6.3

Simplify $\frac{- 12 \pm 2 \sqrt{218}}{2}$ .

$x = - 6 \pm \sqrt{218}$

Step 6.4

Change the $\pm$ to $+$ .

$x = - 6 + \sqrt{218}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 182}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 182$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 182}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 182$ .

$x = \frac{- 12 \pm \sqrt{144 + 728}}{2 \cdot 1}$

Step 7.1.3

Add $144$ and $728$ .

$x = \frac{- 12 \pm \sqrt{872}}{2 \cdot 1}$

Step 7.1.4

Rewrite $872$ as $2^{2} \cdot 218$ .

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Step 7.1.4.1

Factor $4$ out of $872$ .

$x = \frac{- 12 \pm \sqrt{4 (218)}}{2 \cdot 1}$

Step 7.1.4.2

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

$x = \frac{- 12 \pm \sqrt{2^{2} \cdot 218}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{218}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 2 \sqrt{218}}{2}$

Step 7.3

Simplify $\frac{- 12 \pm 2 \sqrt{218}}{2}$ .

$x = - 6 \pm \sqrt{218}$

Step 7.4

Change the $\pm$ to $-$ .

$x = - 6 - \sqrt{218}$

Step 8

The final answer is the combination of both solutions.

$x = - 6 + \sqrt{218}, - 6 - \sqrt{218}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 6 + \sqrt{218}, - 6 - \sqrt{218}$

Decimal Form:

$x = 8.76482306 \dots, - 20.76482306 \dots$