Q10 of 242 Page 8

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

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.

If the roots of the equation (a² + b²)x² - 2(ab + cd)x + (c² + d²) = 0 are equal, prove that .

If the roots of the equations and are simultaneously real, then prove that b² = ac.

If the roots of the equation are equal, prove that either a = 0 or a³ + b³ + c³ = 3abc.

Show that the equation has no real roots, when .