Question

lf $\alpha ,\beta$ are the roots of $4x^2+7x+2=0$, then the equation whose roots are ${\alpha }^2,{\beta }^2$ is

• A. $16x^2-33x+4=0$
• B. $16x^2+33x+4=0$
• C. $4x^2-49x+2=0$
• D. $16x^2-4x+2=0$

Answer 1

The correct option is A $16x^2-33x+4=0$

$Let\alpha ,\beta betherootsofthegivenequation4x^2+7x+2=0\therefore \alpha +\beta =-\frac{7}{4}and\alpha \beta =\frac{2}{4}=\frac{1}{2}Now,{\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta ={(-\frac{7}{4})}^2-2.\frac{1}{2}=\frac{33}{16}....\left(1\right)and{\alpha }^2{\beta }^2={\left(\alpha \beta \right)}^2={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$ ......(2)

$\therefore$ The new equation with ${\alpha }^2,{\beta }^2$ as roots will be

$x^2-({\alpha }^2+{\beta }^2)x+{\alpha }^2{\beta }^2=0$

$\therefore$ Substituting from equation (1) and (2) we have,

$x^2-\frac{33}{16}x+\frac{1}{4}=0\Rightarrow 16x^2-33x+4=0$
Hence, option A is correct.