Question

If $\alpha$ and $\beta$ are zeroes of the polynomial $2x^2+3x+7$. Find a quadratic polynomial whose zeroes are $\frac{1}{{\alpha }^2}$ & $\frac{1}{{\beta }^2}$.

• A. $49x^2+18x+4$
• B. $49x^2-18x+4$
• C. $49x^2-23x-4$
• D. $49x^2-21x+4$

Answer 1

The correct option is A $49x^2+18x+4$

$2x^2+3x+7=0$

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

$\alpha \beta =\frac{7}{2}$

Sum of zeros $=\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

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

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

$=\frac{{\left(\frac{-3}{2}\right)}^2-{(2\times \frac{7}{2})}^2}{{\left(\frac{7}{2}\right)}^2}$

$=\frac{\frac{9}{4}-7}{\frac{49}{4}}$

$=-\frac{18}{49}$

Product of zeros $=\frac{1}{{\alpha }^2{\beta }^2}$

$=\frac{1}{{\left(\frac{7}{2}\right)}^2}$

$=\frac{4}{49}$

$\therefore$ The polynomial is as follows:

$x^2+\left(\frac{18}{49}\right)x+\frac{4}{49}=0$

$\therefore 49x^2+18x+4=0$