Find the Excluded Values 1/(x-2)-5/(x+5)=7/(x^2+3x-10)

$\frac{1}{x - 2} - \frac{5}{x + 5} = \frac{7}{x^{2} + 3 x - 10}$

Step 1

Subtract $\frac{7}{x^{2} + 3 x - 10}$ from both sides of the equation.

$\frac{1}{x - 2} - \frac{5}{x + 5} - \frac{7}{x^{2} + 3 x - 10} = 0$

Step 2

Simplify $\frac{1}{x - 2} - \frac{5}{x + 5} - \frac{7}{x^{2} + 3 x - 10}$ .

Factor $x^{2} + 3 x - 10$ using the AC method.

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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$\frac{1}{x - 2} - \frac{5}{x + 5} - \frac{7}{(x - 2) (x + 5)} = 0$

To write $\frac{1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{1}{x - 2} \cdot \frac{x + 5}{x + 5} - \frac{5}{x + 5} - \frac{7}{(x - 2) (x + 5)} = 0$

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

$\frac{1}{x - 2} \cdot \frac{x + 5}{x + 5} - \frac{5}{x + 5} \cdot \frac{x - 2}{x - 2} - \frac{7}{(x - 2) (x + 5)} = 0$

Write each expression with a common denominator of $(x - 2) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x - 2}$ by $\frac{x + 5}{x + 5}$ .

$\frac{x + 5}{(x - 2) (x + 5)} - \frac{5}{x + 5} \cdot \frac{x - 2}{x - 2} - \frac{7}{(x - 2) (x + 5)} = 0$

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

$\frac{x + 5}{(x - 2) (x + 5)} - \frac{5 (x - 2)}{(x + 5) (x - 2)} - \frac{7}{(x - 2) (x + 5)} = 0$

Reorder the factors of $(x - 2) (x + 5)$ .

$\frac{x + 5}{(x + 5) (x - 2)} - \frac{5 (x - 2)}{(x + 5) (x - 2)} - \frac{7}{(x - 2) (x + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{x + 5 - 5 (x - 2)}{(x + 5) (x - 2)} - \frac{7}{(x - 2) (x + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{x + 5 - 5 (x - 2) - 7}{(x + 5) (x - 2)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x + 5 - 5 x - 5 \cdot - 2 - 7}{(x + 5) (x - 2)} = 0$

Multiply $- 5$ by $- 2$ .

$\frac{x + 5 - 5 x + 10 - 7}{(x + 5) (x - 2)} = 0$

Subtract $5 x$ from $x$ .

$\frac{- 4 x + 5 + 10 - 7}{(x + 5) (x - 2)} = 0$

Add $5$ and $10$ .

$\frac{- 4 x + 15 - 7}{(x + 5) (x - 2)} = 0$

Subtract $7$ from $15$ .

$\frac{- 4 x + 8}{(x + 5) (x - 2)} = 0$

Factor $4$ out of $- 4 x + 8$ .

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) + 8}{(x + 5) (x - 2)} = 0$

Factor $4$ out of $8$ .

$\frac{4 (- x) + 4 (2)}{(x + 5) (x - 2)} = 0$

Factor $4$ out of $4 (- x) + 4 (2)$ .

$\frac{4 (- x + 2)}{(x + 5) (x - 2)} = 0$

Cancel the common factor of $- x + 2$ and $x - 2$ .

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) + 2)}{(x + 5) (x - 2)} = 0$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{4 (- (x) - 1 (- 2))}{(x + 5) (x - 2)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{4 (- (x - 2))}{(x + 5) (x - 2)} = 0$

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{4 (- 1 (x - 2))}{(x + 5) (x - 2)} = 0$

Cancel the common factor.

$\frac{4 (- 1 (x - 2))}{(x + 5) (x - 2)} = 0$

Rewrite the expression.

$\frac{4 \cdot (- 1)}{x + 5} = 0$

Multiply $4$ by $- 1$ .

$\frac{- 4}{x + 5} = 0$

Move the negative in front of the fraction.

$- \frac{4}{x + 5} = 0$

Step 3

Set the denominator in $\frac{4}{x + 5}$ equal to $0$ to find where the expression is undefined.

$x + 5 = 0$

Step 4

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5

Find the Excluded Values 1/(x-2)-5/(x+5)=7/(x^2+3x-10)

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