Pre-Algebra Examples

$(2 x - 11) (x + 51) = 180$

Step 1

Simplify $(2 x - 11) (x + 51)$ .

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Step 1.1

Expand $(2 x - 11) (x + 51)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$2 x (x + 51) - 11 (x + 51) = 180$

Step 1.1.2

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 51 - 11 (x + 51) = 180$

Step 1.1.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 51 - 11 x - 11 \cdot 51 = 180$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.1.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot 51 - 11 x - 11 \cdot 51 = 180$

Step 1.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot 51 - 11 x - 11 \cdot 51 = 180$

Step 1.2.1.2

Multiply $51$ by $2$ .

$2 x^{2} + 102 x - 11 x - 11 \cdot 51 = 180$

Step 1.2.1.3

Multiply $- 11$ by $51$ .

$2 x^{2} + 102 x - 11 x - 561 = 180$

Step 1.2.2

Subtract $11 x$ from $102 x$ .

$2 x^{2} + 91 x - 561 = 180$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $180$ from both sides of the equation.

$2 x^{2} + 91 x - 561 - 180 = 0$

Step 2.2

Subtract $180$ from $- 561$ .

$2 x^{2} + 91 x - 741 = 0$

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 $a = 2$ , $b = 91$ , and $c = - 741$ into the quadratic formula and solve for $x$ .

$\frac{- 91 \pm \sqrt{91^{2} - 4 \cdot (2 \cdot - 741)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 2 \cdot - 741}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 741$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 8 \cdot - 741}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 741$ .

$x = \frac{- 91 \pm \sqrt{8281 + 5928}}{2 \cdot 2}$

Step 5.1.3

Add $8281$ and $5928$ .

$x = \frac{- 91 \pm \sqrt{14209}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 91 \pm \sqrt{14209}}{4}$

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 $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 2 \cdot - 741}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 741$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 8 \cdot - 741}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 741$ .

$x = \frac{- 91 \pm \sqrt{8281 + 5928}}{2 \cdot 2}$

Step 6.1.3

Add $8281$ and $5928$ .

$x = \frac{- 91 \pm \sqrt{14209}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 91 \pm \sqrt{14209}}{4}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 91 + \sqrt{14209}}{4}$

Step 6.4

Rewrite $- 91$ as $- 1 (91)$ .

$x = \frac{- 1 \cdot 91 + \sqrt{14209}}{4}$

Step 6.5

Factor $- 1$ out of $\sqrt{14209}$ .

$x = \frac{- 1 \cdot 91 - 1 (- \sqrt{14209})}{4}$

Step 6.6

Factor $- 1$ out of $- 1 (91) - 1 (- \sqrt{14209})$ .

$x = \frac{- 1 (91 - \sqrt{14209})}{4}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{91 - \sqrt{14209}}{4}$

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 $91$ to the power of $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 4 \cdot 2 \cdot - 741}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 741$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 91 \pm \sqrt{8281 - 8 \cdot - 741}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 741$ .

$x = \frac{- 91 \pm \sqrt{8281 + 5928}}{2 \cdot 2}$

Step 7.1.3

Add $8281$ and $5928$ .

$x = \frac{- 91 \pm \sqrt{14209}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{- 91 \pm \sqrt{14209}}{4}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 91 - \sqrt{14209}}{4}$

Step 7.4

Rewrite $- 91$ as $- 1 (91)$ .

$x = \frac{- 1 \cdot 91 - \sqrt{14209}}{4}$

Step 7.5

Factor $- 1$ out of $- \sqrt{14209}$ .

$x = \frac{- 1 \cdot 91 - (\sqrt{14209})}{4}$

Step 7.6

Factor $- 1$ out of $- 1 (91) - (\sqrt{14209})$ .

$x = \frac{- 1 (91 + \sqrt{14209})}{4}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{91 + \sqrt{14209}}{4}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{91 - \sqrt{14209}}{4}, - \frac{91 + \sqrt{14209}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{91 - \sqrt{14209}}{4}, - \frac{91 + \sqrt{14209}}{4}$

Decimal Form:

$x = 7.05037751 \dots, - 52.55037751 \dots$