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Q.
The equations $x^2 - ax + b = 0$ and $x^2 + bx - a = 0$ have a common root, then
Complex Numbers and Quadratic Equations
Solution:
Let $\alpha$ be a common root of the given equations.
$\therefore \, a^2 - a\alpha + b = 0$ and $a^2 + b\alpha - a= 0$.
$ \Rightarrow \, (a + b) \alpha - (a + b) = 0 $
$\Rightarrow \, (a + b) (\alpha - 1) = 0$
$\Rightarrow \, a + b = 0$ or $\alpha = 1$
If $\alpha = 1$, then $1 - a + b= 0$
$\Rightarrow \, a-b = 1 $.