Question

How to write these questions...........in .......exam.......


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

How to alive these type of questions in exam ...plz give any suggestions...........
plz...plz...

Answer 1

$p(x) = (a-b)x^2+(b-c)x+(c-a)=0 \\ \\ \\ A=(a-b) \\ \\ B=(b-c) \\ \\ C=(c-a)$

 

Roots are equal. So discriminant is zero.    

 

Therefore,  

 

$B^2-4AC = 0 \\ \\ (b-c)^2-(4(a-b)(c-a)) = 0 \\ \\ (b^2-2bc+c^2)-(4(ac-a^2-bc+ab)) = 0 \\ \\ (b^2-2bc+c^2)-(4(ab-bc+ac-a^2)) = 0 \\ \\ (b^2-2bc+c^2)-(4ab-4bc+4ac-4a^2)=0 \\ \\ b^2-2bc+c^2-4ab+4bc-4ac+4a^2=0 \\ \\ 4a^2+b^2+c^2-4ab+2bc-4ac=0 \\ \\ (2a-b-c)^2=0 \\ \\ 2a-b-c=0 \\ \\ 2a=b+c$

 

Hence proved!