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)

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

The equation with roots ${\alpha }^2$ and ${\beta }^2$ can be written as

$x^2-({\alpha }^2+{\beta }^2)x+(\alpha .\beta {)}^2=0$.

$x^2-\left(\right(\alpha +\beta {)}^2-2\alpha .\beta )x+(\alpha .\beta {)}^2=0$.

We know that

$\alpha +\beta =\frac{-b}{a}$ and $\alpha .\beta =\frac{c}{a}$ since

$\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$.

Hence

$x^2-\left(\right(\alpha +\beta {)}^2-2\alpha .\beta )x+(\alpha .\beta {)}^2=0$.

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