Question

(a) Find the value of a so that the equations
$(2a-5)x^2-4x-15=0$
and $(3a-8)x^2-5x-21=0$
have a common root.
(b) IF the equations $x^2-x-p=0$ and $x^2-2xp-12=0$ have common root, find it.
(c) Find the condition on the complex constant $\alpha ,\beta$ if $x^2+\alpha z+\beta =0$ has real roots.

Answer 1

(a) $a=4$ or $8$.
(b) Common root is $2$.
(c) $\overline{z}=z$ as z is real.
Taking conjugate $z^2+\overline{\alpha }z+\overline{\beta }=0$
$\therefore$ $=\frac{z^2}{\alpha \overline{\beta }-\overline{\alpha }\beta }=\frac{z}{\beta -\overline{\beta }}=\frac{1}{\overline{\alpha }-\alpha }$
$\therefore$ ${(\beta -\overline{\beta })}^2=(\overline{\alpha }-\alpha )(\alpha \overline{\beta }-\overline{\alpha }\beta )$