Question

If the quadratic equations $x^2+ax+bc=0$ and $x^2+bx+ac=0$ have a common root then prove that $a+b+c=0$.

Answer 1

Given $x^2+ax+bc=0$ and $x^2+bx+ac=0$ have a common root.

Let $p$ be the common root of the given equations.

Then we've,

$p^2+ap+bc=0$.....(1) and $p^2+pb+ac=0$.....(2).

Now, (2)$-$ (1) we get,

$p(b-a)-c(b-a)=0$

or, $(b-a)(p-c)=0$

or, $p=c$ [ Since $a\ne b$]

Now putting $p=c$ in (1) we get,

$c^2+ac+bc=0$

or, $(c+a+b)=0$ [ Since $c\ne 0$]

or, $a+b+c=0$.