Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Find the equation with roots ${\alpha }^2$ and ${\beta }^2.$

• A. $a(\sqrt{x}{)}^4+b(\sqrt{x}{)}^2+c=0$
• B. $(a\sqrt{x}{)}^2+(b\sqrt{x})+c=0$
• C. $a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$
• D. None of the above

Answer 1

The correct option is C $a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$

Given: $a{x}^2+bx+c=0,a\ne 0$ with roots $\alpha ,\beta$
To find: Transformed quadratic equation with roots ${\alpha }^2$ and ${\beta }^2.$

Let,$t=x^2$

Since, $x=\alpha \Rightarrow t={\alpha }^2$
$x=\beta \Rightarrow t={\beta }^2$

Replace $x\to \sqrt{t}$

$a(\sqrt{t}{)}^2+b(\sqrt{t})+c=0$

In terms of variable x,

$a(\sqrt{x}{)}^2+b(\sqrt{x})+c=0$
This is required equation.