Question

If $\alpha$ and $\beta$ are the zeros of quadratic polynomial $f\left(x\right)=k{x}^2+4x+4$ such that ${\alpha }^2+{\beta }^2=24$, find the value of $k$.

Answer 1

$\alpha ,\beta$ roots of $f\left(x\right)=k{x}^2+4x+4$

Given ${\alpha }^2+{\beta }^2=24$

We know $\alpha +\beta =\frac{-b}{a}=\frac{-4}{k}$

$\alpha \beta =\frac{c}{a}=\frac{4}{k}$

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

${\left(\frac{-4}{k}\right)}^2=24+2\left(\frac{4}{k}\right)$

$\frac{4^2}{k^2}=24+2\left(\frac{4}{k}\right)$

$16=24k^2+8k$
$2=3k^2+k$
$0=3k^2+k-2$
$0=3k^2+3k-2k-2$
$0=3k(k+1)-2(k+1)$
$0=(k+1)(3k-2)$
$\therefore k=-1,\frac{2}{3}$