SOLUTION: Find A, B and C if {{{ (A/(x-1)) + (B/(x-2)) + (C/(x-3)) = (2x^2 - 6x + 6)/((x-1)(x-2)(x-3)) }}}

Algebra ->  Expressions -> SOLUTION: Find A, B and C if {{{ (A/(x-1)) + (B/(x-2)) + (C/(x-3)) = (2x^2 - 6x + 6)/((x-1)(x-2)(x-3)) }}}      Log On


   

Question 1196220: Find A, B and C if
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Multiply both sides by the denominator you see shown on the right-side member.
Simplify the result, and put the left member using the A, B, and C, into the same quadratic form you find shown in your right side member, the quadratic. Now you can compare the parts of the left and right sides.

A%28x-2%29%28x-3%29%2BB%28x-1%29%28x-3%29%2BC%28x-1%29%28x-2%29=2x%5E2-6x%2B6
A%28x%5E2-5x%2B6%29%2BB%28x%5E2-4x%2B3%29%2BC%28x%5E2-3x%2B2%29=2x%5E2-6x%2B6
Ax%5E2-5Ax%2B6A%2BBx%5E2-4Bx%2B3B%2BCx%5E2-3Cx%2B2C
%28A%2BB%2BC%29x%5E2%2B%28-5A-4B-3C%29x%2B%286A%2B3B%2B2C%29=2x%5E2-6x%2B6
%28A%2BB%2BC%29x%5E2-%285A%2B4B%2B3C%29x%2B%286A%2B3B%2B2C%29=2x%5E2-6x%2B6

Comparing corresponding parts,
system%28A%2BB%2BC=2%2C5A%2B4B%2B3C=6%2C6A%2B3B%2B2C=6%29
Solve this system.

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find A, B and C if .
~~~~~~~~~~~~~~~~ 

(1)  Consider both sides at x--> 1.

     Left side tends to infinity as A%2F%28x-1%29. 

     Right side tends to infinity as %282%2A1%5E2-6%2A1%2B6%29%2F%28%28x-1%29%2A%281-2%29%2A%281-3%29%29 = 2%2F%282%2A%28x-1%29%29 = 1%2F%28x-1%29.

     From it, we conclude that A = 1.




(2)  Consider both sides at x--> 2.

     Left side tends to infinity as B%2F%28x-2%29. 

     Right side tends to infinity as %282%2A2%5E2-6%2A2%2B6%29%2F%28%282-1%29%2A%28x-2%29%2A%282-3%29%29 = 2%2F%28%28-1%29%2A%28x-2%29%29 = -2%2F%28x-2%29.

     From it, we conclude that B = -2.




(3)  Consider both sides at x--> 3.

     Left side tends to infinity as C%2F%28x-3%29. 

     Right side tends to infinity as %282%2A3%5E2-6%2A3%2B6%29%2F%28%283-1%29%2A%283-2%29%2A%28x-3%29%29 = 6%2F%282%2A1%2A%28x-3%29%29 = 3%2F%28x-3%29.

     From it, we conclude that C = 3.



ANSWER.  A = 1,  B = -2,  C = 3.

Solved.

It is the way to solve the problem practically "on fingers",
without reducing to the system of 3 equations in 3 unknowns.