Question

If $7\alpha ={\alpha }^2+3$ and ${\beta }^2=7\beta -3,$ then the value of $\alpha \beta$ is

Answer 1

$7\alpha ={\alpha }^2+3$ and ${\beta }^2=7\beta -3$
$\Rightarrow {\alpha }^2-7\alpha +3=0$ and ${\beta }^2-7\beta +3=0$
Clearly, $\alpha ,\beta$ are the roots of $x^2-7x+3=0$
$\therefore$ Product of roots is, $\alpha \beta =\frac{c}{a}=3$