## QRAwarrior 3 years ago 6x + 22 = A(x + 2) + B(x - 4) ^I am doing partial fraction decomposition and this is what I am at. Attempt: "Trick" Let x = -2. Then, 6(2) + 22 = A(-2 + 2) + B(-2 - 4) 12 + 22 = 0 + B(-6) 34 = -6B -34/6 = B ------ "No Trick" 6x + 22 = Ax + 2A + Bx - 4B (LHS = RHS, then coeff's are equal), 6 = A + B, [1] 22 = 2A - 4B => 11 = A - 2B, [2a] [2a] - [1], 5 = 0 -B => -5 = B. Why aren't the two answers the same?

1. satellite73

i am lost. where does this line come from $6x + 22 = Ax + 2A + Bx - 4B$

2. satellite73

oh nvm i see the mistake

3. QRAwarrior

6x + 22/(x+2)(x-4) dx

4. QRAwarrior

What is the mistake?

5. satellite73

Let x = -2. Then, 6(2) + 22 = A(-2 + 2) + B(-2 - 4)

6. satellite73

should be Let x = -2. Then, 6(-2) + 22 = A(-2 + 2) + B(-2 - 4)

7. QRAwarrior

Ok I actually did it with -2, and then this is what I got: 6x + 22 = A(x + 2) + B(x-4). Let x = -2. Then, 6(-2) + 22 = A(-2 + 2) + B(-2-4) -12 + 22 = B(-6) 10/6 = -B -10/6 = B....

8. SUROJ

6x+22=(a+b)x+(2a-4b) now comparing both sides 6=a+b 22=(2a-4b) now solve two equations you will get a and b

9. QRAwarrior

@SUROJ I did that for "No Trick"

10. satellite73

$a+b=6$ $a+2b=11$ $b=5$

11. QRAwarrior

What am I doing wrong wrt the trick? Why am I not getting the same answer as the second method?

12. .Sam.

(6x+22)/((x+2)(x-4)) Factor out the 2, 2(3x+11)/((x+2)(x-4)) ---------------------------------------- A/(x+2)+B/(x-4) ---------------------------------------- 6=A+B 22=-4A+2B ---------------------------------------- A=-5/3 B=23/3 -(5)/(3(x+2))+(23)/(3(x-4))

13. SUROJ

i see mistake when you put 6(2) in place of 6(-2) ....in first trick

14. SUROJ

did you see that?

15. satellite73

ok lets start at the top is it $\frac{6x + 22 }{(x+2)(x-4)}=\frac{a}{x+2}+\frac{b}{x-4}$

16. QRAwarrior

Ok folks, the point here is that the "Trick" does not match with "No Trick" for the below solution: 6x + 22 = A(x + 2) + B(x-4). Let x = -2. Then, 6(-2) + 22 = A(-2 + 2) + B(-2-4) -12 + 22 = B(-6) 10/6 = -B -10/6 = B....

17. QRAwarrior

@satellite73 Yes

18. SUROJ

Let x = -2. Then, 6(2) + 22 = A(-2 + 2) + B(-2 - 4) || in this step 12 + 22 = 0 + B(-6) 34 = -6B -34/6 = B

19. QRAwarrior

@SUROJ Look at my recent one

20. SUROJ

that's right

21. QRAwarrior

Then why is it NOT the same as "No trick"?

22. satellite73

ok so we can do this by writing $6x+22=A(x+2)+B(x-4)$ $-12+22=-6B$ $10=-6B$ $-\frac{5}{3}$ and likewise $A=\frac{23}{3}$ so give $\frac{6x+22}{(x+2)(x-4)}=\frac{23}{3(x+2)}-\frac{5}{3(x+4)}$

23. SUROJ

a+b=6 is same as 2(a+b)=2(6) is same as 2a+2b=12 is same as -2a -2b =-12 add -2a-2b=-12 and 2a-4b=22

24. QRAwarrior

Alright, then how about the "No trick"?

25. satellite73

typo on last line should be $\frac{6x+22}{(x+2)(x-4)}=\frac{23}{3(x+2)}-\frac{5}{3(x-4)}$

26. SUROJ

add left left and right right -2a-2b=-12 and 2a-4b=22 gives -6b=10 so b is -10/6 same as first one

27. satellite73

now we try with "no trick" $6x+22=A(x-4)+B(x+2)$ $6x+22=Ax-4A+Bx+2B$ $6x+22=(A+B)x+2B-4A$ $A+B=6$ $2B-4A=22$

28. satellite73

i get the same answer solving this one

29. SUROJ

yes.....