Problem 604

Solution:
Let $y = x - 5$, so $x = y + 5$.

Then $f(y) = -3(x)^{2} + 45x - 108 = -3(y+5)^{2} + 45(y+5) - 108$.

Expand $(y+5)^{2}$:
$(y+5)^{2} = y^{2} + 10y + 25$

So:
$f(y) = -3(y^{2} + 10y + 25) + 45y + 225 - 108$
$= -3y^{2} - 30y - 75 + 45y + 225 - 108$
$= -3y^{2} + 15y + (225 - 108 - 75)$
$= -3y^{2} + 15y + 42$

So $f(y) = -3y^{2} + 15y + 42$.

The roots of $f(x)$ are the values of $x$ such that $f(x) = 0$:

$-3x^{2} + 15x + 42 = 0$

Divide both sides by $-3$:
$x^{2} - 5x - 14 = 0$

Factor:
$(x - 7)(x + 2) = 0$

So the roots are $x = 7$ and $x = -2$.