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 $2x^2+3x+1=0$
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
$x^2-\left({\alpha }^2+{\beta }^2\right)x+{\alpha }^2{\beta }^2=0$
$\Rightarrow x^2-\frac{5}{4}x+\frac{1}{4}=0$
$\Rightarrow 4x^2-5x+1=0$