Question

If ${}^{\prime }{\alpha }^{\prime }$and ${}^{\prime }{\beta }^{\prime }$ are the roots of quadratic equation $x^2+4x+1=0$, then find the value of
$\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

Answer 1

$x^2+4x+1=0$

$\alpha +\beta =-4$

$\alpha \beta =1$

$\to \frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

$={(\frac{1}{\alpha }+\frac{1}{\beta })}^2-\frac{2}{\alpha \beta }$

$={\left(\frac{\alpha +\beta }{\alpha \beta }\right)}^2-\frac{2}{\alpha \beta }$

$={\left(\frac{-4}{1}\right)}^2-\frac{2}{1}$

$=16-2$

$=14$