Question

Determine the nature of the roots of the following quadratic equation:

(x - 2a)(x - 2b) = 4ab

Answer 1

The given equation is

(x - 2a)(x - 2b) = 4ab

x² - 2bx - 2ax + 4ab = 4ab

x² - 2bx - 2ax + 4ab - 4ab = 0

x² - 2bx - 2ax = 0

x² - 2(a + b)x = 0

The given equation is of the form of ax² + bx + c = 0

where a = 1, b = 2(a + b), c = 0

Therefore, the discriminant

D = b² - 4ac

= (2(a + b))² - 4 × (1) × (0)

= 4(a + b)²

= 4(a² + b² + 2ab)

= 4a² + 4b² + 8ab

∵ D > 0

∴ The roots of the given equation are real and distinct.