Q9 of 242 Page 8

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

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

D = b² – 4ac

If D ≥ 0, roots are real

⇒ 4b² – 4ac ≥ 0

⇒ b² ≥ ac ------ (1)

⇒ 4ac – 4b² ≥ 0

⇒ b² ≤ ac ----- (2)

For both (1) and (2) to be true

⇒ b² = ac

If the roots of the equation are equal, then prove that 2b = a + c.

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

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

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

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