Question

Find condition that the equation $x^2-ax+b=0$ and $x^2+bx-a=0$ have a common root.

Answer 1

Let the common root be $p$ and the other roots are $q,r$

$⟹p+q=a,p+r=-b$

$pq=b,pr=-a$

$\frac{q}{r}=-\frac{b}{a}⟹q=-\frac{b}{a}r$ and $q-r=a+b$

$⟹-\frac{b}{a}r-r=a+b⟹-r(\frac{a+b}{a})=(a+b)⟹r=-a$

$⟹q=b$

$⟹p=a-q=a-b$

substituting $q$ in $x^2-ax+b=0$

$b^2-ab+b=0⟹b=0$ or $a=b+1$