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 $\sqrt{\alpha }$ and $\sqrt{\beta }.$

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

Answer 1

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

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

Let,$t=\sqrt{x}$

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

Replace $x\to t^2$

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

In terms of variable x,

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