Question

If α and β are the roots of the equation x² + px + q = 0 and α⁴ and β⁴ are the roots of the equation x²– rx + s = 0, then the equation x² – 4qx + 2q² – r = 0 has always:

• A. 2 real roots
• B. 2 positive roots
• C. 2 negative roots
• D. 1 positive and 1 negative root

Answer 1

The correct option is D 1 positive and 1 negative root

Given question is flexible. So, we can assume the roots and can make an other quardatic equation.

Let α = 1 and β = 2, then the expression becomes x² – 3x + 2 = 0. So p = -3, q = 2, r = 17, s = 16. The new expression is x² – 8x – 9 = 0. 1 root is positive and other is negative.