Question

Prove that both the roots of the equation(x-a)9X-b)+(X-b)
(X-c)+(x-c) (x-a) are real but they are equal only when .a=b=c

Answer 1

Answer:

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

D = b2 – 4ac

If D > 0, roots are real.

⇒ x2 – (a + b)x + ab + x2 – (b + c)x + bc + x2 – (a + c)x + ac = 0

⇒ 3x2 - 2(a + b + c)x + ab + bc + ac = 0

⇒ D = 4(a + b + c)2 – 12(ab + bc + ac)

⇒ D = a2 + b2 + c2 + 2ab + 2ac + 2bc – 3ab – 3bc – 3ac

⇒ D = 1/2 × (2a2 + 2b2 + 2c2 - 2ab – 2ac – 2bc)

⇒ D = 1/2 × ((a – b)2 + (b – c)2 + (c – a)2)

 

Thus, D is always greater than 0, and the roots are real

Now, when a = b = c,

 

D = 0, thus the roots are equal when a = b = c.