Question

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

• A. $a(a+b)=2bc$
• B. $c(a+c)=2ab$
• C. $b(a+b)=2ac$
• D. $b(a+b)=ac$

Answer 1

Answer: (C)

$b(a+b)=2ac$

Let the roots of given equation be u and v (only two roots because this is a quadratic equation), then

$u+v$ $=u^2+v^2$ (given)

Adding $2uv$ on both the sides,

$u+v+2uv=$ $u^2+v^2$ $+2uv$

$\Rightarrow$ $\frac{-b}{a}$ + $2\frac{c}{a}$ $=(u+v{)}^2$

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

$\Rightarrow$ $b^2$ $+ba=2ac$

$\Rightarrow$ $b(a+b)=2ac$

So. the correct option is 'C'.