# Simplify using partial fractions: 3/(x^2-5x -50).

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The denominator x^2 - 5x - 50 can be factorized as:

x^2 - 5x - 50

=> x^2 - 10x + 5x - 50

=> x(x - 10) + 5(x - 10)

=> (x + 5)(x - 10)

The denominator has the roots x = -5 and x = 10

Now to find the partial fractions of 3/(x^2-5x -50), we can use this method.

The partial fractions are A / (x + 5) + B/(x - 10)

To determine A, cover x + 5 and substitute in 3/(x - 10) the value of the root due to x + 5 , or x = -5

we get 3/(-5 - 10) = -3/15 = -1/5

Similarly for B, cover x - 10 and substitute in 3/(x + 5) the root due to x - 10 or x = 10

we get 3/(5 + 10) = 1/5

**The partial fractions of 3/(x^2-5x -50) are 1/5*(x -10) - 1/5*(x + 5)**

3/(x^2 -5x -5)

We need to simplify as a sum of two fractions.

First let us factor the denominator.

==> x^2 -5x -50 = (x-10)(x+5)

==> 3/(x-10)(x+5) = A/(x-10) + B/(x+5)

We will multiply by (x-10)(x+5)

==> 3 = A(x+5) + B(x-10)

==> 3 = Ax + 5A + Bx -10B

==> 3 = (A+B)x + (5A-10B)

==> A+B = 0 ==> A= -B

==> 5A-10B= 3

==> -5B-10B= 3

==> -15B = 3

==> B= -3/15 = -1/5

==> A= 1/5

**==> 3/(x^2 -5x -15) = 1/5(x-10) - 1/5(x+5)**