If α , β are the real roots of x 2 + p x + q = 0 and α 4 , β 4 are the roots of x 2 - r x + s = 0 , then the equation x 2 - 4 q x + 2 q 2 - r = 0 has always

Step-by-step solution

It is given that, α ,   β are the roots of x 2 + p x + q = 0 α + β = - p and α β = q Since, α 4 ,   β 4 are roots of x 2 - r x + s = 0 Therefore, α 4 + β 4 = r and α 4 β 4 = s Now, x 2 - 4 q x + 2 q 2 - r = 0 D = ( 4 q ) 2 - 4 2 q 2 - r = 16 q 2 - 8 q 2 + 4 r = 8 q 2 + 4 r Here, r = α 4 + β 4 ≥ 0 and 8 q 2 ≥ 0 . Thus, D ≥ 0 Therefore, the equation has two real roots.