Pre-Algebra Examples

$- 2 x (x + 9) - 20 = 10$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot 9 - 20 = 10$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$- 2 (x \cdot x) - 2 x \cdot 9 - 20 = 10$

Step 1.2.2

Multiply $x$ by $x$ .

$- 2 x^{2} - 2 x \cdot 9 - 20 = 10$

Step 1.3

Multiply $9$ by $- 2$ .

$- 2 x^{2} - 18 x - 20 = 10$

Step 2

Subtract $10$ from both sides of the equation.

$- 2 x^{2} - 18 x - 20 - 10 = 0$

Step 3

Subtract $10$ from $- 20$ .

$- 2 x^{2} - 18 x - 30 = 0$

Step 4

Factor $- 2$ out of $- 2 x^{2} - 18 x - 30$ .

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Step 4.1

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

$- 2 x^{2} - 18 x - 30 = 0$

Step 4.2

Factor $- 2$ out of $- 18 x$ .

$- 2 x^{2} - 2 (9 x) - 30 = 0$

Step 4.3

Factor $- 2$ out of $- 30$ .

$- 2 x^{2} - 2 (9 x) - 2 \cdot 15 = 0$

Step 4.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (9 x)$ .

$- 2 (x^{2} + 9 x) - 2 \cdot 15 = 0$

Step 4.5

Factor $- 2$ out of $- 2 (x^{2} + 9 x) - 2 (15)$ .

$- 2 (x^{2} + 9 x + 15) = 0$

Step 5

Divide each term in $- 2 (x^{2} + 9 x + 15) = 0$ by $- 2$ and simplify.

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Step 5.1

Divide each term in $- 2 (x^{2} + 9 x + 15) = 0$ by $- 2$ .

$\frac{- 2 (x^{2} + 9 x + 15)}{- 2} = \frac{0}{- 2}$

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.

$\frac{- 2 (x^{2} + 9 x + 15)}{- 2} = \frac{0}{- 2}$

Step 5.2.1.2

Divide $x^{2} + 9 x + 15$ by $1$ .

$x^{2} + 9 x + 15 = \frac{0}{- 2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 2$ .

$x^{2} + 9 x + 15 = 0$

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 $a = 1$ , $b = 9$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 15)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 15$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 15}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $15$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 1}$

Step 8.1.3

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{21}}{2}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 15$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 15}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $15$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 1}$

Step 9.1.3

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{21}}{2}$

Step 9.3

Change the $\pm$ to $+$ .

$x = \frac{- 9 + \sqrt{21}}{2}$

Step 9.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 + \sqrt{21}}{2}$

Step 9.5

Factor $- 1$ out of $\sqrt{21}$ .

$x = \frac{- 1 \cdot 9 - 1 (- \sqrt{21})}{2}$

Step 9.6

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{21})$ .

$x = \frac{- 1 (9 - \sqrt{21})}{2}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{9 - \sqrt{21}}{2}$

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 $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot 15$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 15}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $15$ .

$x = \frac{- 9 \pm \sqrt{81 - 60}}{2 \cdot 1}$

Step 10.1.3

Subtract $60$ from $81$ .

$x = \frac{- 9 \pm \sqrt{21}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm \sqrt{21}}{2}$

Step 10.3

Change the $\pm$ to $-$ .

$x = \frac{- 9 - \sqrt{21}}{2}$

Step 10.4

Rewrite $- 9$ as $- 1 (9)$ .

$x = \frac{- 1 \cdot 9 - \sqrt{21}}{2}$

Step 10.5

Factor $- 1$ out of $- \sqrt{21}$ .

$x = \frac{- 1 \cdot 9 - (\sqrt{21})}{2}$

Step 10.6

Factor $- 1$ out of $- 1 (9) - (\sqrt{21})$ .

$x = \frac{- 1 (9 + \sqrt{21})}{2}$

Step 10.7

Move the negative in front of the fraction.

$x = - \frac{9 + \sqrt{21}}{2}$

Step 11

The final answer is the combination of both solutions.

$x = - \frac{9 - \sqrt{21}}{2}, - \frac{9 + \sqrt{21}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - \sqrt{21}}{2}, - \frac{9 + \sqrt{21}}{2}$

Decimal Form:

$x = - 2.20871215 \dots, - 6.79128784 \dots$