Pre-Algebra Examples

$2 x^{2} - 4 x + 18 x - 76 = 110$

Step 1

Add $- 4 x$ and $18 x$ .

$2 x^{2} + 14 x - 76 = 110$

Step 2

Subtract $110$ from both sides of the equation.

$2 x^{2} + 14 x - 76 - 110 = 0$

Step 3

Subtract $110$ from $- 76$ .

$2 x^{2} + 14 x - 186 = 0$

Step 4

Factor $2$ out of $2 x^{2} + 14 x - 186$ .

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Step 4.1

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

$2 (x^{2}) + 14 x - 186 = 0$

Step 4.2

Factor $2$ out of $14 x$ .

$2 (x^{2}) + 2 (7 x) - 186 = 0$

Step 4.3

Factor $2$ out of $- 186$ .

$2 x^{2} + 2 (7 x) + 2 \cdot - 93 = 0$

Step 4.4

Factor $2$ out of $2 x^{2} + 2 (7 x)$ .

$2 (x^{2} + 7 x) + 2 \cdot - 93 = 0$

Step 4.5

Factor $2$ out of $2 (x^{2} + 7 x) + 2 \cdot - 93$ .

$2 (x^{2} + 7 x - 93) = 0$

Step 5

Divide each term in $2 (x^{2} + 7 x - 93) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (x^{2} + 7 x - 93) = 0$ by $2$ .

$\frac{2 (x^{2} + 7 x - 93)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} + 7 x - 93)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $x^{2} + 7 x - 93$ by $1$ .

$x^{2} + 7 x - 93 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$x^{2} + 7 x - 93 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = 7$ , and $c = - 93$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 93)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 93}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 93$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 93}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 93$ .

$x = \frac{- 7 \pm \sqrt{49 + 372}}{2 \cdot 1}$

Step 8.1.3

Add $49$ and $372$ .

$x = \frac{- 7 \pm \sqrt{421}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{421}}{2}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 93}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 93$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 93}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 93$ .

$x = \frac{- 7 \pm \sqrt{49 + 372}}{2 \cdot 1}$

Step 9.1.3

Add $49$ and $372$ .

$x = \frac{- 7 \pm \sqrt{421}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{421}}{2}$

Step 9.3

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{421}}{2}$

Step 9.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{421}}{2}$

Step 9.5

Factor $- 1$ out of $\sqrt{421}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{421})}{2}$

Step 9.6

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{421})$ .

$x = \frac{- 1 (7 - \sqrt{421})}{2}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{421}}{2}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 93}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 93$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 93}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 93$ .

$x = \frac{- 7 \pm \sqrt{49 + 372}}{2 \cdot 1}$

Step 10.1.3

Add $49$ and $372$ .

$x = \frac{- 7 \pm \sqrt{421}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{421}}{2}$

Step 10.3

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{421}}{2}$

Step 10.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{421}}{2}$

Step 10.5

Factor $- 1$ out of $- \sqrt{421}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{421})}{2}$

Step 10.6

Factor $- 1$ out of $- 1 (7) - (\sqrt{421})$ .

$x = \frac{- 1 (7 + \sqrt{421})}{2}$

Step 10.7

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{421}}{2}$

Step 11

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{421}}{2}, - \frac{7 + \sqrt{421}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{421}}{2}, - \frac{7 + \sqrt{421}}{2}$

Decimal Form:

$x = 6.75914226 \dots, - 13.75914226 \dots$