Question

Let $a,b,c$ be real numbers with $a\ne 0$ and let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$. Express the roots of $a^3{x}^2+abcx+c^3=0$ in terms of $\alpha ,\beta$.

Answer 1

Let $f\left(x\right)=a{x}^2+bx+c$

$\Rightarrow f\left(\frac{a}{c}x\right)=\frac{a^3{x}^2}{c^2}+\frac{bax+c}{c}$

$=\frac{a^3{x}^2+abcx+c^3}{c^2}$

$f\left(x\right)=0$ when $x=\alpha ,x=\beta$

$\Rightarrow f\left(\frac{a}{c}x\right)=0$ when $\frac{a}{c}x=\alpha ,\frac{a}{c}x=\beta$

$\therefore$ Roots of $a^3{x}^2+abcx+c^3=0$ are

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