Question

If two roots of the quadratic equation (b-c)x^2+(c-a)x+(a-b)=0 are equal
,then let us prove that,2b=a+c

Answer 1

Rule :

Let us take an equation

ax² + bx + c = 0 .....(i)

If the roots of (i) be equal, the value of its discriminant be 0. Then

b² - 4ac = 0

Solution :

The given equation is

(b - c)x² + (c - a)x + (a - b) = 0

For equal roots, we must have

(c - a)² - 4 (b - c) (a - b) = 0

or, c² - 2ca + a² - 4ab + 4b² + 4ca - 4bc = 0

or, 4b² + a² + c² - 4ab - 4bc + 2ca = 0

or, (2b - a - c)² = 0

Thus, 2b - a - c = 0 = 2b - a - c

So, 2b - a - c = 0

==> 2b = a + c

Hence, proved.

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Answer 2

plz refer to this attachment