Browse Questions

# The roots of $ax^2+bx+c=0$ are $r$ and $s$. For the roots of $x^2+px+q=0$ to be $r^2$ and $s^2$, what must be the value of $p$ ?

(A) $\large\frac{b^2-4ac}{a^2}$
(B) $\large\frac{b^2-2ac}{a^2}$
(C) $\large\frac{2ac-b^2}{a^2}$
(D) $b^2-2c$

Toolbox:
• Sum of the roots of $ax^2+bx+c=0$ is $\large-\frac{b}{a}$
• Product of the roots = $\large\frac{c}{a}$
• $a^2+b^2=(a+b)^2-2ab$
• Quadratic equation whose roots are $\alpha\:\:and\:\:\beta$ is $x^2-(\alpha+\beta)x+\alpha\beta$
Given: $r\:\:and\:\:s$ are roots of $ax^2+bx+c=o$
$\Rightarrow\:r+s=\large-\frac{b}{a}$ and $rs=\large\frac{c}{a}$
$r^2+s^2=(r+s)^2-2rs$
$=\large\frac{b^2}{a^2}-$$2\large\frac{c}{a}$$=\large\frac{b^2-2ac}{a^2}$
$r^2s^2=\large\frac{c^2}{a^2}$
Quadratic Eqn., whose roots are $r^2\:\:and\:\:s^2$ is
$x^2-(r^2+s^2)x+r^2s^2$
$=x^2-(\large\frac{b^2-2ac}{a^2})$$x+\large\frac{c^2}{a^2}$
Given: $x^2+px+q=0$ is equation whose roots are $r^2\:and\:s^2$
$\Rightarrow\:p=-\large\frac{b^2-2ac}{a^2}=\frac{2ac-b^2}{a^2}$