Question

If $\alpha$ and $\beta$ are the roots of the equation $ax ^2+ bx + c =0$ then the sum of the roots of the equation $a^2 x^2+\left(b^2-2 a c\right) x+b^2-4 a c=0$ in terms of $\alpha$ and $\beta$ is given by

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

Answer 1

Answer: (D)

$\alpha+\beta=-\frac{ b }{ a } ; \alpha \beta=\frac{ c }{ a } ; x _1+ x _2=\frac{2 ac - b ^2}{ a ^2}=\frac{2 c }{ a }-\left(\frac{ b }{ a }\right)^2=2 \alpha \beta-(\alpha+\beta)^2=-\left(\alpha^2+\beta^2\right.$