Question

Prove that the roots of the following equation are rational
$(a+c-b)x^2+2cx+(b+c-a=0)$

Answer 1

If discriminant is a perfect square then roots are rational.

$D=b^2-4ac$

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

$D=4c^2-4[ab+ac-a^2+bc+c^2-ac-b^2-bc+ab]$

$D=4c^2-4[-a^2-b^2+c^2+2ab]$

$D=4c^2-4c^2+4(a^2+b^2-2ab)$

$D=4(a-b{)}^2$

$D=\left(2\right(a-b){)}^2$

$D$ is perfect square of $2(a-b)$

Hence roots are rational.