Question

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

Answer 1

If the quadratic equation ax²+bx+c=0

whose roots are equal then it's

deteminant is equal to zero.

(a-b)x²+(b-c)x+(c-a)=0

Deteminant =0

(b-c)² -4(a-b)(c-a)==0

b²+c²-2bc-4ac+4a²+4bc-4ab=0

b²+c²+4a²+4bc-4ac-4ab=0

b²+c²+(-2a)²+2bc+2c(-2a)+2(-2a)b=0

(b+c-2a)²=0

b+c-2a=0

Therefore,

b+c=2a

Hence proved.