Question

If $\alpha$ and $\beta$ are roots of $a{x}^2+bx+c=0$ then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

• A. $a^2{x}^2-(b^2-2ac)x+c^2=0$
• B. $a^2{x}^2+(b^2-ac)x+c^2=0$
• C. $a^2{x}^2+(b^2+ac)x+c^2=0$
• D. $a^2{x}^2+(b^2+2ac)x+c^2=0$

Answer 1

Answer: (A)

Let $y=x^2\Rightarrow x=\sqrt{y}$
putting $\sqrt{y}$ in the given equation
$ay+b\sqrt{y}+c=0\Rightarrow b\sqrt{y}=-ay-c\Rightarrow b^{2}y=a^2{y}^2+c^2+2acy$
$\Rightarrow a^2{y}^2-\left(b^2-2ac\right)y+c^2=0$
So the required quadratic equation is $a^2{x}^2-\left(b^2-2ac\right)x+c^2=0$