Question

Partial fraction decomposition
20x+9/25x^2+20x+4

Answer 1

Explanation:

We have

`\frac{20x + 9}{25 {x}^(2) + 20x + 4 }`

Let's factor the denomiator first,

the denomaitor is a perfect square so we get

`\frac{20x + 9}{(5x + 2) {}^(2) }`

Now, we must think of two fractions that

`\frac{a}{(5x + 2) {}^(2) } + (b)/(5x + 2)`

We use a perfect square term for one fraction, then a linear one for the next, because if we set both of the denomiator to the same factor, we would get a inconsistent system.

So right now, we have

`\frac{a}{(5x + 2) { }^(2) } + (b)/(5x + 2) = \frac{20x +9 }{25 {x}^(2) + 20x + 4 }`

`a + (5x + 2)b = 20x + 9`

`5b (x)= 20`

`a + 2b = 9`

`b = 4`

so that means that a is

`a + (2)(4) = 9`

`a + 8 = 9`

`a = 1`

So our equation is

`\frac{1}{(5x + 2) {}^(2) } + (4)/(5x + 2)`