If p, q are real and p ≠ q, then show that the roots of the equation are real and unequal

Question

If p, q are real and p &#x2260; q, then show that the roots of the equation are real and unequal

Philoid · Accepted Answer

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D > 0, roots are real and unequal.

⇒ D = 25(p + q)² + 8(p – q)²

Thus D > 0 for all p and q as sum of two squares is always positive.