Question

The equation whose roots are the squares of the roots of the equation $2x ^{2}+3x +1 =0$ is

• A. $4x ^{2}+5x +1 =0$
• B. $4x ^{2}-x +1 =0$
• C. $4x ^{2}-5x -1 =0$
• D. $4x ^{2}-5x +1 =0$
• E. $4x ^{2}+5x -1 =0$

Answer 1

Answer: (D)

Let the roots of the equation $2 x^{2}+3 x+1=0$
are $\alpha$ and $\beta$.
Then, $ \alpha+\beta=\frac{-3}{2}\,\,\,...(i)$
and $ \alpha \beta =\frac{1}{2}\,\,\,...(ii)$
$\therefore \alpha^{2}+\beta^{2} =(\alpha+\beta)^{2}-2 \alpha \beta$
$=\left(\frac{-3}{2}\right)^{2}-2\left(\frac{1}{2}\right)=\frac{9}{4}-1=\frac{5}{4}$
and $\alpha^{2} \beta^{2}=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$
$\therefore $ Required equation is
$x^{2}-\left(\alpha^{2}+\beta^{2}\right) x+\alpha^{2} \beta^{2}=0 $
$ \Rightarrow x^{2}-\frac{5}{4} x+\frac{1}{4}=0$
$\Rightarrow 4 x^{2}-5 x+1=0 $