# Decompose into partial fractions : (5x + 10) /x ( x +5)

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We have to write (5x + 10) / x*( x +5) as sum of partial fractions.

(5x + 10) / x*( x +5)

=> A / x + B / (x + 5)

=> (5A + Ax + xB)/x*(x + 5)

This gives 5A + Ax + xB = 5x + 10

equate the terms with x and the numeric terms

5A = 10

=> A = 2

A + B = 5

=> B = 3

**The expression as partial fractions is 2/x + 3/(x + 5)**

We need to write as partial fractions as follow:

(5x+10)/x(x+5) = A/x + B/(x+5)

We will multiply by x(x+5) both sides.

==> A(x+5) + Bx = 5x + 10

Now we will group similar terms.

==> Ax + 5A + Bx = 5x + 10

==> (A+B)x + 5A = 5x + 10

Now we will compare;

==> 5A = 10 ==> A = 2

==> A+B = 5 ==> B = 5 - A = 5-2= 3

==> B = 3

Now we will substitute into the equation.

**==> (5x+10)/x(x+5) = (2/x) + 3/(x+5)**

We'll create irreducible elementary fractions:

(5x + 10) /x ( x +5) = A/x + B/(x+5)

(5x + 10) = A(x+5) + B*x

(5x + 10) = Ax + 5A + Bx

(5x + 10) = x(A+B) + 5A

Comparing, we'll get:

A+B = 5

5A=10 => A = 2

2+B = 5 => B = 3

**The result of decomposing into partial fractions is:**

**(5x + 10) /x ( x +5) = 2/x + 3/(x+5)**