Algebra Examples

$\frac{6}{x + 3} - \frac{5}{x - 2} = \frac{- 20}{x^{2} + x - 6}$

Step 1

Subtract $\frac{- 20}{x^{2} + x - 6}$ from both sides of the equation.

$\frac{6}{x + 3} - \frac{5}{x - 2} - \frac{- 20}{x^{2} + x - 6} = 0$

Step 2

Simplify $\frac{6}{x + 3} - \frac{5}{x - 2} - \frac{- 20}{x^{2} + x - 6}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $x^{2} + x - 6$ using the AC method.

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Step 2.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 2.1.1.2

Write the factored form using these integers.

$\frac{6}{x + 3} - \frac{5}{x - 2} - \frac{- 20}{(x - 2) (x + 3)} = 0$

Step 2.1.2

Move the negative in front of the fraction.

$\frac{6}{x + 3} - \frac{5}{x - 2} - - \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.1.3

Multiply $- - \frac{20}{(x - 2) (x + 3)}$ .

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Step 2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{6}{x + 3} - \frac{5}{x - 2} + 1 \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.1.3.2

Multiply $\frac{20}{(x - 2) (x + 3)}$ by $1$ .

$\frac{6}{x + 3} - \frac{5}{x - 2} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.2

To write $\frac{6}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{6}{x + 3} \cdot \frac{x - 2}{x - 2} - \frac{5}{x - 2} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.3

To write $- \frac{5}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{6}{x + 3} \cdot \frac{x - 2}{x - 2} - \frac{5}{x - 2} \cdot \frac{x + 3}{x + 3} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.4

Write each expression with a common denominator of $(x + 3) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{6}{x + 3}$ by $\frac{x - 2}{x - 2}$ .

$\frac{6 (x - 2)}{(x + 3) (x - 2)} - \frac{5}{x - 2} \cdot \frac{x + 3}{x + 3} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.4.2

Multiply $\frac{5}{x - 2}$ by $\frac{x + 3}{x + 3}$ .

$\frac{6 (x - 2)}{(x + 3) (x - 2)} - \frac{5 (x + 3)}{(x - 2) (x + 3)} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.4.3

Reorder the factors of $(x - 2) (x + 3)$ .

$\frac{6 (x - 2)}{(x + 3) (x - 2)} - \frac{5 (x + 3)}{(x + 3) (x - 2)} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{6 (x - 2) - 5 (x + 3)}{(x + 3) (x - 2)} + \frac{20}{(x - 2) (x + 3)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{6 (x - 2) - 5 (x + 3) + 20}{(x + 3) (x - 2)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{6 x + 6 \cdot - 2 - 5 (x + 3) + 20}{(x + 3) (x - 2)} = 0$

Step 2.7.2

Multiply $6$ by $- 2$ .

$\frac{6 x - 12 - 5 (x + 3) + 20}{(x + 3) (x - 2)} = 0$

Step 2.7.3

Apply the distributive property.

$\frac{6 x - 12 - 5 x - 5 \cdot 3 + 20}{(x + 3) (x - 2)} = 0$

Step 2.7.4

Multiply $- 5$ by $3$ .

$\frac{6 x - 12 - 5 x - 15 + 20}{(x + 3) (x - 2)} = 0$

Step 2.8

Subtract $5 x$ from $6 x$ .

$\frac{x - 12 - 15 + 20}{(x + 3) (x - 2)} = 0$

Step 2.9

Subtract $15$ from $- 12$ .

$\frac{x - 27 + 20}{(x + 3) (x - 2)} = 0$

Step 2.10

Add $- 27$ and $20$ .

$\frac{x - 7}{(x + 3) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$x - 7 = 0$

Step 4

Add $7$ to both sides of the equation.

$x = 7$