Question

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

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

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 $a^{2}x^{2}-\left(b^{2}-2ac\right)x+c^{2}=0$