Question

The equation whose roots are the squares of the roots of the equation $a{x}^2+bx+c=0$ is

• A. $a^2{x}^2+b^{2}x+c^2=0$
• B. $a^2{x}^2-(b^2-4ac)x+c^2=0$
• C. $a^2{x}^2-(b^2-2ac)x+c^2=0$
• D. $a^2{x}^2+(b^2-ac)x+c^2=0$

Answer 1

Answer: (C)

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

Let $\alpha ,\beta$ be the roots of the given quadratic equation
$a{x}^2+bx+c=0$
$\alpha +\beta =-\frac{b}{a}$
$\alpha \beta =\frac{c}{a}$
Now, sum of roots $={\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
$=\frac{b^2-2ac}{a^2}$
Product of roots $={\alpha }^2{\beta }^2$
$=\frac{c^2}{a^2}$
Required quadratic equation is
$x^2-\frac{b^2-2ac}{a^2}x+\frac{c^2}{a^2}=0$
$\Rightarrow a^2{x}^2-(b^2-2ac)x+c^2=0$