If the roots of the quadratic equation (a – b)x² + (b – c)x + (c – a) = 0 are equal, prove that 2a = b + c.
well, this one's easy, if the equation is of the form Ax^2 +Bx +C=0 then the roots are -B/2A and -B/2A
so the roots are x= -b+c/(2a-2b) and x = -b+c/(2a-2b)
we equate the roots to 1 then we can prove that 2a = b + c but then, how 1 come from ( I GUESSED TO EQUATE TO 1), well, please help
Method 1: (School Method,Straightforward)
If the roots are equal, then the discriminant is 0.
So
Method 2: (Simpler, Elegant)
We see that 1 is a root of the equation since
So if the roots are equal, the other root also must be 1. This means the product of the roots is 1. Thus