Question

lf the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to sum of their squares, then

• A. $ab+b^2+2ac=0$
• B. $ab+a^2+2ac=0$
• C. $ab+b^2-2ac=0$
• D. $ab+a^2-2ac=0$

Answer 1

The correct option is C $ab+b^2-2ac=0$

Let $\alpha ,\beta$ are roots of $a{x}^2+bx+c=0$, then

$\alpha +\beta =-\frac{b}{a}$

$\alpha \beta =\frac{c}{a}$

As sum of roots is equal to sum of their square, then

$\alpha +\beta ={\alpha }^2+{\beta }^2$

$\Rightarrow \alpha +\beta ={(\alpha +\beta )}^2-2\alpha \beta$

$\Rightarrow -\frac{b}{a}={(-\frac{b}{a})}^2-2\left(\frac{c}{a}\right)$

$\Rightarrow -\frac{b}{a}=\frac{b^2}{a^2}-\frac{2c}{a}$

$\Rightarrow -ab=b^2-2ac$

$\Rightarrow b^2+ab-2ac=0$