Simplify (x^2-7x+10)/(x^2-2x-15)+(x-2)/(3x-15)

$\frac{x^{2} - 7 x + 10}{x^{2} - 2 x - 15} + \frac{x - 2}{3 x - 15}$

Step 1

Simplify each term.

Factor $x^{2} - 7 x + 10$ 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 $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$\frac{(x - 5) (x - 2)}{x^{2} - 2 x - 15} + \frac{x - 2}{3 x - 15}$

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$\frac{(x - 5) (x - 2)}{(x - 5) (x + 3)} + \frac{x - 2}{3 x - 15}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) (x - 2)}{(x - 5) (x + 3)} + \frac{x - 2}{3 x - 15}$

Rewrite the expression.

$\frac{x - 2}{x + 3} + \frac{x - 2}{3 x - 15}$

Factor $3$ out of $3 x - 15$ .

Factor $3$ out of $3 x$ .

$\frac{x - 2}{x + 3} + \frac{x - 2}{3 (x) - 15}$

Factor $3$ out of $- 15$ .

$\frac{x - 2}{x + 3} + \frac{x - 2}{3 x + 3 \cdot - 5}$

Factor $3$ out of $3 x + 3 \cdot - 5$ .

$\frac{x - 2}{x + 3} + \frac{x - 2}{3 (x - 5)}$

Step 2

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

$\frac{x - 2}{x + 3} \cdot \frac{3 (x - 5)}{3 (x - 5)} + \frac{x - 2}{3 (x - 5)}$

Step 3

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

$\frac{x - 2}{x + 3} \cdot \frac{3 (x - 5)}{3 (x - 5)} + \frac{x - 2}{3 (x - 5)} \cdot \frac{x + 3}{x + 3}$

Step 4

Write each expression with a common denominator of $(x + 3) \cdot 3 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{(x - 2) (3 (x - 5))}{(x + 3) (3 (x - 5))} + \frac{x - 2}{3 (x - 5)} \cdot \frac{x + 3}{x + 3}$

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

$\frac{(x - 2) (3 (x - 5))}{(x + 3) (3 (x - 5))} + \frac{(x - 2) (x + 3)}{3 (x - 5) (x + 3)}$

Reorder the factors of $(x + 3) (3 (x - 5))$ .

$\frac{(x - 2) (3 (x - 5))}{3 (x + 3) (x - 5)} + \frac{(x - 2) (x + 3)}{3 (x - 5) (x + 3)}$

Reorder the factors of $3 (x - 5) (x + 3)$ .

$\frac{(x - 2) (3 (x - 5))}{3 (x + 3) (x - 5)} + \frac{(x - 2) (x + 3)}{3 (x + 3) (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 2) (3 (x - 5)) + (x - 2) (x + 3)}{3 (x + 3) (x - 5)}$

Step 6

Simplify the numerator.

Factor $x - 2$ out of $(x - 2) \cdot 3 (x - 5) + (x - 2) (x + 3)$ .

Factor $x - 2$ out of $(x - 2) \cdot 3 (x - 5)$ .

$\frac{(x - 2) (3 (x - 5)) + (x - 2) (x + 3)}{3 (x + 3) (x - 5)}$

Factor $x - 2$ out of $(x - 2) (3 (x - 5)) + (x - 2) (x + 3)$ .

$\frac{(x - 2) (3 (x - 5) + x + 3)}{3 (x + 3) (x - 5)}$

Apply the distributive property.

$\frac{(x - 2) (3 x + 3 \cdot - 5 + x + 3)}{3 (x + 3) (x - 5)}$

Multiply $3$ by $- 5$ .

$\frac{(x - 2) (3 x - 15 + x + 3)}{3 (x + 3) (x - 5)}$

Add $3 x$ and $x$ .

$\frac{(x - 2) (4 x - 15 + 3)}{3 (x + 3) (x - 5)}$

Add $- 15$ and $3$ .

$\frac{(x - 2) (4 x - 12)}{3 (x + 3) (x - 5)}$

Factor $4$ out of $4 x - 12$ .

Factor $4$ out of $4 x$ .

$\frac{(x - 2) (4 (x) - 12)}{3 (x + 3) (x - 5)}$

Factor $4$ out of $- 12$ .

$\frac{(x - 2) (4 x + 4 \cdot - 3)}{3 (x + 3) (x - 5)}$

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$\frac{(x - 2) (4 (x - 3))}{3 (x + 3) (x - 5)}$

$\frac{(x - 2) \cdot 4 (x - 3)}{3 (x + 3) (x - 5)}$

Step 7

Move $4$ to the left of $x - 2$ .

$\frac{4 (x - 2) (x - 3)}{3 (x + 3) (x - 5)}$

Simplify (x^2-7x+10)/(x^2-2x-15)+(x-2)/(3x-15)

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