Question

If $\alpha$ and $\beta$ are zeros of the polynomial $2x^2-4x+5$, find the values of:
$\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

• A. $4$
• B. $-\frac{4}{25}$
• C. $25$
• D. None of these

Answer 1

Answer: (B)

$-\frac{4}{25}$

$\left(\frac{1}{\alpha }\right)+\left(\frac{1}{\beta }\right)=\left(\frac{{\beta }^2+{\alpha }^2}{{\alpha }^2{\beta }^2}\right)=\left(\frac{(\alpha +\beta {)}^2-2\alpha \beta }{{\alpha }^2{\beta }^2}\right)=\left(\frac{(\frac{4}{2}{)}^2-2(\left(\frac{5}{2}\right))}{(\frac{5}{2}{)}^2}\right)=\left(\frac{4-5}{\left(\frac{25}{4}\right)}\right)=\left(\frac{-4}{25}\right)$