How do you express #\frac { 5x - 6} { x ( x - 3) }# in the form #A/x + B/(x-3)#?

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Answer 1 · 2017-02-01T20:37:59

#2/x+3/(x-3)#

First, try to put the form into the original like this:

#A/x+B/(x-3)=(A(x-3))/(x(x-3))+(Bx)/(x(x-3))=(A(x-3)+Bx)/(x(x-3))#
#=((A+B)x-3A)/(x(x-3))#

Now this last form is very similar to the original problem

#(5x-6)/(x(x-3))#

In this case

#A+B=5# and #3A=6#

Solving this system of two equations in two unknowns gives:

#A=2# and #B=3#

Finally, we can plug #A=2# and #B=3# back into the form we were asked to provide:

#2/x+3/(x-3)#