Question

If sum of the roots is $p$ and the sum of their squares is $q^2$, the quadratic equation is

• A. $x^2-px+p^2{q}^3=0$
• B. $x^2-px+q^3=0$
• C. $x^2-px+\frac{p^2-q^2}{2}=0$
• D. None of these

Answer 1

The correct option is C $x^2-px+\frac{p^2-q^2}{2}=0$

Let $\alpha$ and $\beta$ be the roots of the equation.
As per the given condition, $\alpha +\beta =p$ and ${\alpha }^2+{\beta }^2=q^2$
We know, ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
$\Rightarrow q^2=p^2-2\alpha \beta$

$\Rightarrow 2\alpha \beta =p^2-q^2$
$\Rightarrow \alpha \beta =\frac{p^2-q^2}{2}$
Now sum of the roots is $p$ and product of the roots is $\frac{p^2-q^2}{2}$

So the equation is

$x^2-px+\frac{p^2-q^2}{2}=0$

Hence, option C is correct.