Question

If $\alpha ,\beta$ are the roots of $2x^2-3x+4=0$, then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

• A. $4x^2+7x+16=0$
• B. $4x^2-7x-16=0$
• C. $4x^2-7x+16=0$
• D. $4x^2+7x-16=0$

Answer 1

The correct option is A $4x^2+7x+16=0$

Let $f\left(x\right)=2x^2-3x+4=0\to$ roots are $\alpha ,\beta$
The equation whose roots are ${\alpha }^2,{\beta }^2$ is $f\left(\sqrt{x}\right)=0$
$\Rightarrow 2x-3\sqrt{x}+4=0\Rightarrow 2x+4=3\sqrt{x}$
$\Rightarrow (2x+4{)}^2=9x\Rightarrow 4x^2+7x+16=0$