Simpify into two partial fractions:2x-3/ (x^2 -5x -50)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the partial fractions of 2x-3/ (x^2 -5x -50)

2x-3/ (x^2 -5x -50)

=> 2x - 3 / (x^2 - 10x + 5x - 50)

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

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

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

=> Ax - 10A + Bx + 5B = 2x - 3

=> A + B  = 2 and 10A - 5B = 3

=> 10A + 10B = 20 and 10A - 5B = 3

subtract the two equations

=> 15B = 17

=> B = 17/15

A = 13/15

The partial fractions are 13/15*(x + 5) + 17/15*(x - 10)

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

First we will factor the denominator.

==> (x^2 -5x -50) = (x-10)(x+5) = 0

Then the partial fractions are:

(2x-3)/(x-10)(x+5) = A/(x-10) + B/(x+5)

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

==> 2x-3 = A(x+5) + B(x-10)

==> 2x-3 = (A+B)x + 5A-10B

==> A+B = 2 ==> A = 2-B...........(1)

==> 5(A-2B) = -3

==> 5( 2-B -2B) = -3

==> 10 - 15B = -3

==> 15B = 13

==> B= 13/15

==> A= 2-B= 2 - 13/15 = (30-13)/15 = 17/15

==> (2x-3)/x^2 - 5x -50 = 17/15(x-10) + 13/15(x+5)

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