Question

Find the quadratic equation such that itsroots are square of sum of roots & square of difference of the roots of equation.2 x² + 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