Question

Find quadratic equation such that its roots are square of sum of the roots and square of difference of the roots of equation 2x² + 2 (p + q) x+ p²+ q²= 0

Answer 1

Let the roots of the required quation be M and N
let the roots of the equation 2x²+2(p+q)x+p²+q²=0 be a and b
a + b = -(p+q)
ab = (p^2 + q^2) / 2
(a+b)^2 = (p+q)^2
(a-b)^2 = (a+b)^2 - 4ab
(a-b)^2 = -(p - q)^2
we wanted the values of square of sum of the roots and square of difference of the roots
Now M = (a+b)^2 = (p+q)^2 and
N = (a-b)^2 = -(p - q)^2
M + N = 4pq
MN = (p+q)^2 [-(p - q)^2]
MN= -(p^2 - q^2)^2
hence the required equation is
x^2 - (4pq)x - (p^2 - q^2)^2 = 0
Hope this helps!!!