Simplify 1/(x-3)+4/(x^2-2x-3)

$\frac{1}{x - 3} + \frac{4}{x^{2} - 2 x - 3}$

Step 1

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$\frac{1}{x - 3} + \frac{4}{(x - 3) (x + 1)}$

Step 2

To write $\frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{1}{x - 3} \cdot \frac{x + 1}{x + 1} + \frac{4}{(x - 3) (x + 1)}$

Step 3

Multiply $\frac{1}{x - 3}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x + 1}{(x - 3) (x + 1)} + \frac{4}{(x - 3) (x + 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x + 1 + 4}{(x - 3) (x + 1)}$

Step 5

Add $1$ and $4$ .

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

Simplify 1/(x-3)+4/(x^2-2x-3)

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