Question

If $\alpha$ and $\beta$ are roots of $a{x}^2+bx+c=0$, the find the quadratic equation which has ${\alpha }^2+2$ and ${\beta }^2+2$ as roots.

Answer 1

$a{x}^2+bx+c=0$

sum of roots, $\alpha +\beta =\frac{-b}{a}$

product of roots $\Rightarrow \alpha \beta =\frac{c}{a}$

$\Rightarrow$ Since $e{q}^n$ has roots as ${\alpha }^2+2$ & ${\beta }^2+2$

So sum $\Rightarrow {\alpha }^2+2+{\beta }^2+2$

$={\alpha }^2+{\beta }^2+4$

$=(\alpha +\beta {)}^2-2\alpha \beta +4$

$=(\frac{-b}{a}{)}^2-2(\frac{c}{a})+4$

$=\frac{+b^2-2ac+4a^2}{a^2}$

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

$={\alpha }^2{\beta }^2+2({\alpha }^2+{\beta }^2)+4$

$=(\frac{\alpha }{a}{)}^2+2[(\alpha +\beta {)}^2-2\alpha \beta ]+4$

$=\frac{c^2}{a^2}+2\left[\right(\frac{-b}{a}{)}^2-2\left(\frac{c}{a}\right)]+4$

$=\frac{c^2}{a^2}+2(\frac{b^2}{a^2}-\frac{2c}{a})+4$

$=\frac{c^2+2b^2-4ac+4a^2}{a^2}$

$\therefore$ The quadratic $e{q}^n$ will be :-

$\Rightarrow x^2+x\left[\frac{-(b^2-ac+4a^2)}{a^2}\right]+\frac{c^2+2b^2-4ac+4a^2}{a^2}$

$\Rightarrow a^2{x}^2+x(ac-b^2-4a^2)+(c^2+2b^2-4ac+4a^2)=0$