Question

if the roots of the equation (a-b)x^2 + (b-c)x +(c-a)=0 are equal prove that 2a= b+c please solve it in simple method

Answer 1

● Quadratic equations ●

Given equation : - (a - b)² + (b - c)x + (c - a)

The roots of the equation are equal.

To prove : - 2a = b + c.

That means, the discriminant = D = B² - 4AC = 0

So,here,

B = (b - c), A = (a - b) and C = (c - a)

B² - 4AC = 0

=> (b - c)² - 4 × (a - b) × (c - a) = 0

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

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

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

=> (b + c)² - 4ab - 4bc + 4a² = 0

=> (b + c)² - 4a(b + c) + 4a² = 0

=> (b + c)² - 2a(b + c) - 2a(b + c) + 4a² = 0

=> (b + c)[(b + c) - 2a] - 2a[(b + c) - 2a] = 0

=> (b + c - 2a)(b + c - 2a) = 0

=> (b + c - 2a)² = 0

=> b + c - 2a = 0

=> b + c = 2a

=> 2a = b + c

Hence, proved.