# Write the ratio (3x+2)/(x^2+x) using partial fraction decomposition.

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We have to write (3x+2)/(x^2+x) as sum of fractions.

Now (x^2 + x )= x ( x +1)

(3x+2)/(x^2+x)

=> A / (x+1) + B/x

=> (Ax + Bx + B)/ x(x+1)

So we get A + B = 3 and B = 2

As B = 2 , A = 3 - 2 = 1

So the required form is 1/(x+1) + 2/x.

**Therefore (3x+2)/(x^2+x) = 1/(x+1) + 2/x.**

To decompose a fraction we'll have to factor the denominator. We'll factorize by x:

x^2+x = x( x + 1 )

We'll write the fractions with one of the factors for each of the denominators.

(3x+2)/(x^2+x) = A/x + B/( x + 1 )

We'll multiply both side by the common denominator x( x + 1 ):

3x+2 = A(x+1) + Bx

We'll remove the brackets:

3x+2 = Ax + A + Bx

We'll factorize by x to the right side:

3x+2 = x(A+B) + A

We'll compare and we'll get:

A+B = 3

A = 2

2 + B = 3

B = 3 - 2

B = 1

**So, the fraction (3x+2)/(x^2+x) = 2/x + 1/(x + 1)**