Question

If the roots of the quadratic equation $(a+c-2b)x^2+(b+a-2c)x+(b+c-2a)=0$ are equal then find a condition on $a,b,c$.

Answer 1

Given the roots quadratic eqation

$(a+c-2b)x^2+(b+a-2c)x+(b+c-2a)=0$

are equal.

Then, $D=0$

$⟹(b+a-2c{)}^2=4(b+c-2a\left)\right(a+c-2b)$

or, $(b^2+a^2+4c^2+2ab-4bc-4ac)$

$=4\{ab+bc-2b^2+ac+c^2-2bc-2a^2-2ac+4ab\}$

or, $9a^2+9b^2+2ab-20ab=0$

or, $a^2+b^2-2ab=0$

or, $(a-b{)}^2=0$

or, $a=b$.

So, The required condition is $a=b$.