Question

If a and b are real and a ≠ b then show that the roots of the equation (a - b)x² + 5(a + b)x - 2(a - b) = 0. are equal and unequal.

Answer 1

the given equation is (a - b)x² + 5(a + b)x - 2(a - b) = 0

∴ D = [5(a + b)]² - 4 x (a - b) x [-2(a - b)]

= 25(a + b)² + 8(a - b)²

Since a and b are real and a ≠ b, so (a - b)² > 0 and (a + b)² > 0

∴8(a - b)² > 0 ......... (1) (Product of two positive numbers is always positive)

Also, 25(a + b)² > 0 .......(2) (Product of two positive numbers is always positive)

Adding (1) and (2), we get

25(a + b)² + 8(a - b)² > 0 (Sum of two positive numbers is always positive)

⇒ D > 0

Hence, the roots of the given equation are real and unequal.