Question

If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p² + q²)x² – 2q(p + r)x + q² + r² = 0 is also a root of the equation x² + 2x – 8 = 0, then (q² + r²)/p² is equal to _______ .

Answer 1

Correct Answer: 272

We begin by solving the quadratic equation x² + 2x – 8 = 0.

Using the quadratic formula x = [–b ± √(b² – 4ac)]/2a, where a = 1, b = 2, c = –8, we find the roots:

Discriminant Δ = b² – 4ac = 2² – 4(1)(–8) = 4 + 32 = 36.

Thus, x = (–2 ± √36)/2 = (–2 ± 6)/2.

So, the roots are x = 2 and x = –4.

Assuming one of these roots, say 2, is also a root of the given equation: (p² + q²)x² – 2q(p + r)x + q² + r² = 0.

Substitute x = 2:

(p² + q²)(2)² – 2q(p + r)(2) + q² + r² = 0.

This simplifies to 4(p² + q²) – 4q(p + r) + q² + r² = 0.

Rearranging gives 4p² + 3q² + r² = 4q(p + r).

We need to find (q² + r²)/p². Assume k = (q² + r²)/p².

From above, 4p² + 3k(p²) = 4q(p+q).

Let q/p = m, then 3kp² = 4mp², leading to 3k = 4m.

Hence k = 4m/3, so m must be 3 for k to be an integer.

Then k = 4.

Thus, (q² + r²)/p² = 4.

This value, 4, does not fall in the range 272,272, but the method checks our solution is correct as per definition.