How do you use the rational root theorem to find the roots of ##P(x) = 0.25x^2 - 12x + 23##?

##x=2" "## or ##" "x=46##

First multiply by ##4## to make all of the coefficients into integers:

##4P(x) = 4(0.25x^2-12x+23) = x^2-48x+92##

By the rational root theorem, any of ##x^2-48x+92## (and therefore of ##P(x)##) are expressible in the form ##p/q## for integers ##p, q## with ##p## a divisor of the constant term ##92## and ##q## a divisor of the coefficient ##1## of the leading term.

That means that the only possible rational are:

##+-1, +-2, +-4, +-23, +-46, +-92##

Trying each in turn we soon find:

##P(color(blue)(2)) = 0.25(color(blue)(2))^2-12(color(blue)(2))+23 = 1-24+23 = 0##

So ##x=2## is a zero and ##(x-2)## a factor:

##x^2-48x+92 = (x-2)(x-46)##

So the other zero of ##P(x)## is ##x=46##