Question

Where did I go wrong?

Answer 1

?

Answer 2

a, b and c are all different numbers in AP. If the roots of ax^2 + bx + c = 0 are p and q and 1/p + 1/q, p + q, p^2 + q^2 is in GP, then find a/c.

So I started by simplifying the GP to

p + q/pq, p + q, p^2 + q^2
=> -b/c, -b/a, (b^2 - 2ac)/a^2

And then I imposed the condition that

-b/a * c/a = (b^2 - 2ac)/a^2
=> -bc = b^2 - 2ac
=> b^2 + bc - 2ac = 0

Now since a, b, c are in AP, I substituted b = (a+c)/2

=> [(a+c)/2]^2 + c * [(a+c)/2] - 2ac = 0
=> (a^2 + c^2 + 2ac)/4 + (ac + c^2)/2 - 2ac = 0
=> a^2 + c^2 + 2ac + 2ac + 2c^2 - 8ac = 0
=> a^2 + 3c^2 - 4ac = 0
=> (a/c)^2 + 3 - 4(a/c) = 0
=> (a/c)^2 - 3(a/c) - (a/c) + 3 = 0
=> a/c = 1, 3
We eliminate 1.