Precalculus Examples

$\frac{- 10 x + 20}{x^{2} - x - 20} x = - 3$

Step 1

Factor each term.

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Step 1.1

Multiply $\frac{- 10 x + 20}{x^{2} - x - 20}$ by $x$ .

$\frac{- 10 x + 20}{x^{2} - x - 20} x = - 3$

Step 1.2

Combine $\frac{- 10 x + 20}{x^{2} - x - 20}$ and $x$ .

$\frac{(- 10 x + 20) x}{x^{2} - x - 20} = - 3$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2} - x - 20, 1$

Step 2.2

Remove parentheses.

$x^{2} - x - 20, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{2} - x - 20$

Step 3

Multiply each term in $\frac{(- 10 x + 20) x}{x^{2} - x - 20} = - 3$ by $x^{2} - x - 20$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{(- 10 x + 20) x}{x^{2} - x - 20} = - 3$ by $x^{2} - x - 20$ .

$\frac{(- 10 x + 20) x}{x^{2} - x - 20} (x^{2} - x - 20) = - 3 (x^{2} - x - 20)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{2} - x - 20$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(- 10 x + 20) x}{x^{2} - x - 20} (x^{2} - x - 20) = - 3 (x^{2} - x - 20)$

Step 3.2.1.2

Rewrite the expression.

$(- 10 x + 20) x = - 3 (x^{2} - x - 20)$

Step 3.2.2

Apply the distributive property.

$- 10 x \cdot x + 20 x = - 3 (x^{2} - x - 20)$

Step 3.2.3

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

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Step 3.2.3.1

Move $x$ .

$- 10 (x \cdot x) + 20 x = - 3 (x^{2} - x - 20)$

Step 3.2.3.2

Multiply $x$ by $x$ .

$- 10 x^{2} + 20 x = - 3 (x^{2} - x - 20)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$- 10 x^{2} + 20 x = - 3 x^{2} - 3 (- x) - 3 \cdot - 20$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $- 1$ by $- 3$ .

$- 10 x^{2} + 20 x = - 3 x^{2} + 3 x - 3 \cdot - 20$

Step 3.3.2.2

Multiply $- 3$ by $- 20$ .

$- 10 x^{2} + 20 x = - 3 x^{2} + 3 x + 60$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Add $3 x^{2}$ to both sides of the equation.

$- 10 x^{2} + 20 x + 3 x^{2} = 3 x + 60$

Step 4.1.2

Subtract $3 x$ from both sides of the equation.

$- 10 x^{2} + 20 x + 3 x^{2} - 3 x = 60$

Step 4.1.3

Add $- 10 x^{2}$ and $3 x^{2}$ .

$- 7 x^{2} + 20 x - 3 x = 60$

Step 4.1.4

Subtract $3 x$ from $20 x$ .

$- 7 x^{2} + 17 x = 60$

Step 4.2

Subtract $60$ from both sides of the equation.

$- 7 x^{2} + 17 x - 60 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = - 7$ , $b = 17$ , and $c = - 60$ into the quadratic formula and solve for $x$ .

$\frac{- 17 \pm \sqrt{17^{2} - 4 \cdot (- 7 \cdot - 60)}}{2 \cdot - 7}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot - 7 \cdot - 60}}{2 \cdot - 7}$

Step 4.5.1.2

Multiply $- 4 \cdot - 7 \cdot - 60$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 17 \pm \sqrt{289 + 28 \cdot - 60}}{2 \cdot - 7}$

Step 4.5.1.2.2

Multiply $28$ by $- 60$ .

$x = \frac{- 17 \pm \sqrt{289 - 1680}}{2 \cdot - 7}$

Step 4.5.1.3

Subtract $1680$ from $289$ .

$x = \frac{- 17 \pm \sqrt{- 1391}}{2 \cdot - 7}$

Step 4.5.1.4

Rewrite $- 1391$ as $- 1 (1391)$ .

$x = \frac{- 17 \pm \sqrt{- 1 \cdot 1391}}{2 \cdot - 7}$

Step 4.5.1.5

Rewrite $\sqrt{- 1 (1391)}$ as $\sqrt{- 1} \cdot \sqrt{1391}$ .

$x = \frac{- 17 \pm \sqrt{- 1} \cdot \sqrt{1391}}{2 \cdot - 7}$

Step 4.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{2 \cdot - 7}$

Step 4.5.2

Multiply $2$ by $- 7$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{- 14}$

Step 4.5.3

Simplify $\frac{- 17 \pm i \sqrt{1391}}{- 14}$ .

$x = \frac{17 \pm i \sqrt{1391}}{14}$

Step 4.6

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

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot - 7 \cdot - 60}}{2 \cdot - 7}$

Step 4.6.1.2

Multiply $- 4 \cdot - 7 \cdot - 60$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 17 \pm \sqrt{289 + 28 \cdot - 60}}{2 \cdot - 7}$

Step 4.6.1.2.2

Multiply $28$ by $- 60$ .

$x = \frac{- 17 \pm \sqrt{289 - 1680}}{2 \cdot - 7}$

Step 4.6.1.3

Subtract $1680$ from $289$ .

$x = \frac{- 17 \pm \sqrt{- 1391}}{2 \cdot - 7}$

Step 4.6.1.4

Rewrite $- 1391$ as $- 1 (1391)$ .

$x = \frac{- 17 \pm \sqrt{- 1 \cdot 1391}}{2 \cdot - 7}$

Step 4.6.1.5

Rewrite $\sqrt{- 1 (1391)}$ as $\sqrt{- 1} \cdot \sqrt{1391}$ .

$x = \frac{- 17 \pm \sqrt{- 1} \cdot \sqrt{1391}}{2 \cdot - 7}$

Step 4.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{2 \cdot - 7}$

Step 4.6.2

Multiply $2$ by $- 7$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{- 14}$

Step 4.6.3

Simplify $\frac{- 17 \pm i \sqrt{1391}}{- 14}$ .

$x = \frac{17 \pm i \sqrt{1391}}{14}$

Step 4.6.4

Change the $\pm$ to $+$ .

$x = \frac{17 + i \sqrt{1391}}{14}$

Step 4.7

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

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot - 7 \cdot - 60}}{2 \cdot - 7}$

Step 4.7.1.2

Multiply $- 4 \cdot - 7 \cdot - 60$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 17 \pm \sqrt{289 + 28 \cdot - 60}}{2 \cdot - 7}$

Step 4.7.1.2.2

Multiply $28$ by $- 60$ .

$x = \frac{- 17 \pm \sqrt{289 - 1680}}{2 \cdot - 7}$

Step 4.7.1.3

Subtract $1680$ from $289$ .

$x = \frac{- 17 \pm \sqrt{- 1391}}{2 \cdot - 7}$

Step 4.7.1.4

Rewrite $- 1391$ as $- 1 (1391)$ .

$x = \frac{- 17 \pm \sqrt{- 1 \cdot 1391}}{2 \cdot - 7}$

Step 4.7.1.5

Rewrite $\sqrt{- 1 (1391)}$ as $\sqrt{- 1} \cdot \sqrt{1391}$ .

$x = \frac{- 17 \pm \sqrt{- 1} \cdot \sqrt{1391}}{2 \cdot - 7}$

Step 4.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{2 \cdot - 7}$

Step 4.7.2

Multiply $2$ by $- 7$ .

$x = \frac{- 17 \pm i \sqrt{1391}}{- 14}$

Step 4.7.3

Simplify $\frac{- 17 \pm i \sqrt{1391}}{- 14}$ .

$x = \frac{17 \pm i \sqrt{1391}}{14}$

Step 4.7.4

Change the $\pm$ to $-$ .

$x = \frac{17 - i \sqrt{1391}}{14}$

Step 4.8

The final answer is the combination of both solutions.

$x = \frac{17 + i \sqrt{1391}}{14}, \frac{17 - i \sqrt{1391}}{14}$