Question

How do you express `(5x - 4 ) / (x^2 -4x)` in partial fractions?

Answer 1

`(5x - 4)/(x^2 - 4x) = 4/(x - 4) + 1/x`

Explanation:

The denominator can be written as `x(x - 4)`.

We can rewrite this as follows:

`A/(x - 4) + B/x = (5x- 4)/(x^2 - 4x)`

Put on a common denominator.

`A(x) + B(x - 4) = 5x - 4`

`Ax + Bx - 4B = 5x - 4`

`(A + B)x - 4B = 5x - 4`

We can hence write the following system of equations.

`A + B = 5`
`-4B = -4`

Solving, we have that `B = 1` and that `A = 4`

We can now reinsert into the original equation to get our partial fraction decomposition.

`4/(x- 4) + 1/x = (5x - 4)/(x^2 - 4x)`

Let's quickly check the validity of our answer.

What is the sum of `4/(x - 4) + 1/x`?

Put on a common denominator.

`(4(x))/((x - 4)(x)) + (1(x- 4))/(x(x - 4)) = (4x + x - 4)/(x(x - 4)) = (5x - 4)/(x^2 - 4x)`

This verifies the initial expression, so we have done our decomposition properly.

Hopefully this helps!