Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $25x^2+25x+5$, then find the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}.$

Answer 1

Given

$25x^2+25x+5$

$\alpha$ and $\beta$ the zeros of the polynomial

$\alpha +\beta =\frac{-b}{a}=\frac{-25}{25}=-1$

$\alpha \beta =\frac{c}{a}=\frac{5}{25}=\frac{1}{5}$

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

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

${\alpha }^2+{\beta }^2+\frac{2}{5}=1$

${\alpha }^2+{\beta }^2=1-\frac{2}{5}=\frac{3}{5}$

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

$=\frac{\left(\frac{3}{5}\right)}{(\frac{1}{5}{)}^2}$

$=\frac{\frac{3}{5}}{\frac{1}{25}}$

$=\frac{3}{5}\times 25=15$