Algebra Examples

$\frac{x + 7}{x + 5} - \frac{7 x}{x^{2} - 3 x - 40}$

Step 1

Factor $x^{2} - 3 x - 40$ using the AC method.

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Step 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 $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 1.2

Write the factored form using these integers.

$\frac{x + 7}{x + 5} - \frac{7 x}{(x - 8) (x + 5)}$

Step 2

To write $\frac{x + 7}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{x + 7}{x + 5} \cdot \frac{x - 8}{x - 8} - \frac{7 x}{(x - 8) (x + 5)}$

Step 3

Write each expression with a common denominator of $(x + 5) (x - 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + 7}{x + 5}$ by $\frac{x - 8}{x - 8}$ .

$\frac{(x + 7) (x - 8)}{(x + 5) (x - 8)} - \frac{7 x}{(x - 8) (x + 5)}$

Step 3.2

Reorder the factors of $(x - 8) (x + 5)$ .

$\frac{(x + 7) (x - 8)}{(x + 5) (x - 8)} - \frac{7 x}{(x + 5) (x - 8)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 7) (x - 8) - 7 x}{(x + 5) (x - 8)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + 7) (x - 8)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{x (x - 8) + 7 (x - 8) - 7 x}{(x + 5) (x - 8)}$

Step 5.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 8 + 7 (x - 8) - 7 x}{(x + 5) (x - 8)}$

Step 5.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 8 + 7 x + 7 \cdot - 8 - 7 x}{(x + 5) (x - 8)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 8 + 7 x + 7 \cdot - 8 - 7 x}{(x + 5) (x - 8)}$

Step 5.2.1.2

Move $- 8$ to the left of $x$ .

$\frac{x^{2} - 8 \cdot x + 7 x + 7 \cdot - 8 - 7 x}{(x + 5) (x - 8)}$

Step 5.2.1.3

Multiply $7$ by $- 8$ .

$\frac{x^{2} - 8 x + 7 x - 56 - 7 x}{(x + 5) (x - 8)}$

Step 5.2.2

Add $- 8 x$ and $7 x$ .

$\frac{x^{2} - x - 56 - 7 x}{(x + 5) (x - 8)}$

Step 5.3

Subtract $7 x$ from $- x$ .

$\frac{x^{2} - 8 x - 56}{(x + 5) (x - 8)}$