The Student Room Group

Prove that for any positive numbers p and q: p + q = sqrt(4pq)

I did (p + q)^2 = p^2 + 2pq + q^2 g 4pq
so p^2 -2qr + q^2 0 which is true since it's equal to (p-q)^2 and any number squared is 0.
Since (p - q)^2 + 4pq = (p + q)^2 and (p - q)^2 0
(p + q)^2 4pq so
p + q sqrt (4 pq)

The solution bank added the line p and q are both positive
so p > 0 and q > 0
Therefore, p + q > 0

Why is this necessary? Doesn't the line "...(p-q)^2 and any number squared is 0 " eliminate the need for it, since it shows that 4qr is the value that (p +q)^2 is equal or greater than?
Reply 1
*bump* whoops
Original post by EmRep13
I did (p + q)^2 = p^2 + 2pq + q^2 g 4pq
so p^2 -2qr + q^2 0 which is true since it's equal to (p-q)^2 and any number squared is 0.
Since (p - q)^2 + 4pq = (p + q)^2 and (p - q)^2 0
(p + q)^2 4pq so
p + q sqrt (4 pq)

The solution bank added the line p and q are both positive
so p > 0 and q > 0
Therefore, p + q > 0

Why is this necessary? Doesn't the line "...(p-q)^2 and any number squared is 0 " eliminate the need for it, since it shows that 4qr is the value that (p +q)^2 is equal or greater than?

Nowhere in your proof do you use that p & q are positive....
Reply 3
Original post by zetamcfc
Nowhere in your proof do you use that p & q are positive....

I don't see why you'd need to? Other than in the last bit just to state that it only applies if p + q are positive? But is that grounds to do so? Do you need to use it if the question states?
Original post by EmRep13
*bump* whoops

When they square rooted both sides of the inequality (p+q)^2 >= 4pq they used the fact that p,q are both positive (a,b>0 a^2>b^2 <==> a>b)) so you need to state this @EmRep13
Original post by EmRep13
I don't see why you'd need to? Other than in the last bit just to state that it only applies if p + q are positive? But is that grounds to do so? Do you need to use it if the question states?

(2)2>(1)2(-2)^2 > (-1)^2 but 2<1-2 < -1, the problem comes when you go from (p+q)24pq(p + q)^2 \ge 4pq to p+q4pqp + q \ge \sqrt {4 pq}. (some constraint on pp, qq is required for this inequality to even make sense)

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