Pre-Algebra Examples

$\frac{- 5}{x - 3} + \frac{3 x - 19}{x^{2} - 8 x + 15}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Move the negative in front of the fraction.

$- \frac{5}{x - 3} + \frac{3 x - 19}{x^{2} - 8 x + 15}$

Step 1.2

Factor $x^{2} - 8 x + 15$ using the AC method.

Tap for more steps...

Step 1.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.2.2

Write the factored form using these integers.

$- \frac{5}{x - 3} + \frac{3 x - 19}{(x - 5) (x - 3)}$

Step 2

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

$- \frac{5}{x - 3} \cdot \frac{x - 5}{x - 5} + \frac{3 x - 19}{(x - 5) (x - 3)}$

Step 3

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

Tap for more steps...

Step 3.1

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

$- \frac{5 (x - 5)}{(x - 3) (x - 5)} + \frac{3 x - 19}{(x - 5) (x - 3)}$

Step 3.2

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

$- \frac{5 (x - 5)}{(x - 3) (x - 5)} + \frac{3 x - 19}{(x - 3) (x - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 5 (x - 5) + 3 x - 19}{(x - 3) (x - 5)}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{- 5 x - 5 \cdot - 5 + 3 x - 19}{(x - 3) (x - 5)}$

Step 5.2

Multiply $- 5$ by $- 5$ .

$\frac{- 5 x + 25 + 3 x - 19}{(x - 3) (x - 5)}$

Step 5.3

Add $- 5 x$ and $3 x$ .

$\frac{- 2 x + 25 - 19}{(x - 3) (x - 5)}$

Step 5.4

Subtract $19$ from $25$ .

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

Step 5.5

Factor $2$ out of $- 2 x + 6$ .

Tap for more steps...

Step 5.5.1

Factor $2$ out of $- 2 x$ .

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

Step 5.5.2

Factor $2$ out of $6$ .

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

Step 5.5.3

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

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

Step 6

Cancel the common factor of $- x + 3$ and $x - 3$ .

Tap for more steps...

Step 6.1

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

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

Step 6.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{2 (- (x) - 1 (- 3))}{(x - 3) (x - 5)}$

Step 6.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 6.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{2 (- 1 (x - 3))}{(x - 3) (x - 5)}$

Step 6.5

Cancel the common factor.

$\frac{2 (- 1 (x - 3))}{(x - 3) (x - 5)}$

Step 6.6

Rewrite the expression.

$\frac{2 \cdot (- 1)}{x - 5}$

Step 7

Multiply $2$ by $- 1$ .

$\frac{- 2}{x - 5}$

Step 8

Move the negative in front of the fraction.

$- \frac{2}{x - 5}$