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

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

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.

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

Step 2

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

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

Step 3

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Add $2 x$ and $x$ .

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

Add $- 1$ and $1$ .

$\frac{3 x + 0}{(x - 3) (x + 1)}$

Add $3 x$ and $0$ .

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

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

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