Question

if $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$ then the sum of the roots of the equation $a^2{x}^2+(b^2-2ac)x+b^2-4ac=0$ is

• A. $-({\alpha }^2-{\beta }^2)$
• B. ${(\alpha +\beta )}^2-2\alpha \beta$
• C. ${\alpha }^2\beta +\beta {\alpha }^2-4\alpha \beta$
• D. $-({\alpha }^2+{\beta }^2)$

Answer 1

The correct option is D $-({\alpha }^2+{\beta }^2)$

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

then

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

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

The sum of the roots of the equation is given by,

sum of roots=

$-\frac{b^2-2ac}{a^2}=-(\frac{b}{a}{)}^2+2\frac{c}{a}⟹=-(\alpha +\beta {)}^2+2\alpha \beta ⟹=-\left({\alpha }^2\right)-{\beta }^2-2\alpha \beta +2\alpha \beta ⟹-({\alpha }^2+{\beta }^2)$