# How do you express 1/ (x^2 + 3x -4) in partial fractions?

Feb 11, 2016

$\frac{1}{5} \left(x - 1\right) - \frac{1}{5} \left(x + 4\right)$

#### Explanation:

first step is to factor the denominator :

 x^2 + 3x - 4 ) = (x+4)(x-1)

since these factors are linear the the numerators of the partial fractions will be constants , say A and B.

$\Rightarrow \frac{1}{\left(x + 4\right) \left(x - 1\right)} = \frac{A}{x + 4} + \frac{B}{x - 1}$

now multiply both sides by )x+4)(x-1)

so 1 = A(x-1) + B(x+4).......................................(1)

The aim now is to find the values of A and B. Note that if x = 1 , the term with A will be zero and if x = -4 the term with B will be zero.
This is the starting point for finding A and B.

let x = 1 in (1) : 1 = 5B $\Rightarrow B = \frac{1}{5}$

let x = -4 in (1) : 1 = -5A $\Rightarrow A = - \frac{1}{5}$

$\Rightarrow \frac{1}{{x}^{2} + 3 x - 4} = \frac{1}{5} \left(x - 1\right) - \frac{1}{5} \left(x + 4\right)$