If p&q = p^2 + q^2 – 2pq, for what value of q is p&

Gmat_mission wrote:If p&q = p^2 + q^2 – 2pq, for what value of q is p&q equal to p^2 for all values of p?

A. -2
B. -1
C. 0
D. 1
E. 2

[spoiler]OA=C[/spoiler].

Is there a strategic approach to solve this PS question? May someone helps me? I'd be thankful.

Hi Gmat_mission.

Here, we have to find the value of p such that $$p\&q=p^2\ \ \ \forall\ p.$$ Which is the same as solving: $$p^2+q^2-2pq=p^2$$ $$\Rightarrow\ \ \ q^2-2pq=0$$ $$\Rightarrow\ \ \ q\left(q-2p\right)=0$$ $$\Rightarrow\ \ \ q=0\ \ \ or\ \ \ \ \ q-2p=0$$ $$\Rightarrow\ \ \ q=0\ \ \ or\ \ \ \ \ q=2p.$$ Since we want to find a value of q such that the equation holds for any value of p, then we have to pick q=0.

Therefore, the answer here is the option [spoiler]C=0[/spoiler].

I hope this answer may help you.

Gmat_mission wrote:If p&q = p^2 + q^2 – 2pq, for what value of q is p&q equal to p^2 for all values of p?

A. -2
B. -1
C. 0
D. 1
E. 2

We see that if q = 0, then p&0 = p^2 - 0^2 - 2p(0) = p^2. So q must be 0.

Alternate solution:

Notice that p^2 + q^2 - 2pq = (p - q)^2, so p&q = (p - q)^2. We see that if q = 0, then (p - q)^2 = p^2.

Answer: C