Question 45
The operations below can be applied on any expression of the form $a x^{2}+b x+c$.
(I) If $c \neq 0$, replace $a$ by $4 a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$.
(II) If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2 c+\frac{3}{a}$.
(III$_{t}$) Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.)
Is it possible to transform $x^{2}-x-6$ into each of the following by applying some sequence of the above operations?
a. $5 x^{2}+5 x-1$
b. $x^{2}+6 x+2$
(I) If $c \neq 0$, replace $a$ by $4 a-\frac{3}{c}$ and $c$ by $\frac{c}{4}$.
(II) If $a \neq 0$, replace $a$ by $-\frac{a}{2}$ and $c$ by $-2 c+\frac{3}{a}$.
(III$_{t}$) Replace $x$ by $x-t$, where $t$ is an integer. (Different values of $t$ can be used.)
Is it possible to transform $x^{2}-x-6$ into each of the following by applying some sequence of the above operations?
a. $5 x^{2}+5 x-1$
b. $x^{2}+6 x+2$