Decompose expression into a sum of partial fractions `(x-17)/(x^2 -9x+14)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression `(x - 17)/(x^2 - 9x + 14)` has to be converted to the partial fractions form.

`(x - 17)/(x^2 - 9x + 14)`

= `(x - 17)/(x^2 - 7x - 2x + 14)`

= `(x - 17)/(x(x - 7) - 2(x - 7))`

= `(x - 17)/((x - 2)(x - 7))`

`(x - 17)/((x - 2)(x - 7)) = A/(x - 2) + B/(x - 7)`

=> `(x - 17)/((x - 2)(x - 7)) = (A(x - 7))/((x - 2)(x-7)) + (B(x - 2))/((x - 7)(x - 2))`


=> `(x - 17)= A(x - 7) + B(x - 2)`

=> x - 17 = Ax - 7A + Bx - 2B

=> 1 = A + B and -17 = -7A - 2B

Substitute A = 1 - B in 7A + 2B = 17

=> 7 - 7B + 2B = 17

=> -5B = 10

=> B = -2

A = 3

The expression `(x - 17)/(x^2 - 9x + 14) = 3/(x - 2) - 2/(x - 7)`

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