arithmetic question

C

I plugged in numbers for u,v, calculated a,b,c , then plugged in the answers.

u=4
v=2

a= 2

b= 12

c= 56

a) 36 + 8 not equal 4*56
b) 12+ 16 not equal 2*56
c) 3*144 + 16 = 8*56 = 448

Night reader wrote:If u-v=a, u^2-v^2=b and u^3-v^3=c, find the equation of a, b and c

A. 3b+2a^2=2ac
B. b+4a^2=2c
C. 3b^2+a^4=4ac
D. 3b^2+a^2=4ac
E. b+4a^2=4c

Hi there!

Please note that this question stem should in fact be "which of the following equations relates a, b and c ?" , therefore we have to look at the alternatives to "focus" on what we should do!

(A) 3b+2a^2 has no terms in u^4 but 2ac has, therefore it is not alternative (A).

(B) b+4a^2 has no terms in u^3 but 2c has, therefore it is not alternative (B).

(C) Please wait till the end.

(D) Left side has a "u*v" term, right side has not, therefore it is not alternative (D).

(E) Left side has no u^3 term, right side has, therefore it is not alternative (E).

In the GMAT, I would go for (C) by exclusion, but now we are studying for the GMAT, so let´s check it:

3b^2 + a^4 = 3(u-v)^2*(u+v)^2 + (u-v)^4 =
= (u-v)^2 *[3(u+v)^2+(u-v)^2] = ... = (u-v)^2*4*(u^2+v^2+uv)

Now please remember that (u-v)(u^2+v^2+uv) = u^3-v^3 what motivates me to do:

(u-v)^2*4*(u^2+v^2+uv) = 4(u-v)(u-v)(u^2+v^2+uv) = 4(u-v)(u^3-v^3) = 4ac , and we proved (C) is really correct.

Best Regards,
Fabio.