If the roots of the quadratic equation (a - b)x2 + (b - c)x + (c - a) = 0 are equal, prove that 2a = b + c.

Question

Philoid · Accepted Answer

We know that roots of a quadratic equation in the form Ax² + Bx + C = 0 are equal if

Discriminant, D = 0

B² - 4AC = 0

In given equation, A = a - b , B = b - c, C = c - a

So, (b - c)² - 4(a - b)(c - a) = 0

b² + c² - 2bc - 4(ac - a² - bc + ab) = 0

b² + c² - 2bc - 4ac + 4a² + 4bc - 4ab = 0

(- 2a)² + b² + c² + 2(- 2a)b + 2bc + 2( - 2a)(c) = 0

(- 2a + b + c )² = 0

- 2a + b + c = 0

b + c = 2a

Hence Proved.

If the roots of the quadratic equation (a - b)x² + (b - c)x + (c - a) = 0 are equal, prove that 2a = b + c.