For what values of $b$ is the expression $x^2-3x+b$ factorable?

Question

This is from a summer work packet for high school AP Calculus. I've never seen anything like this. I played around with it and got this as a best guess: $$\{ b \in \Bbb{Q} \; | \; b \le 2.25 \}$$ I'm not sure if that's formatted the right way but whatever. Sorry in advanced if I am breaking any rules for asking questions (I just skimmed through the "How to Ask" page).

(My bad about copying the wrong problem)

Answer 1

For the expression to be factorable (over the real numbers), the quadratic equation $$ x^2 - 3x + b = 0 $$ has to have two real (not necessarily distinct) roots. This condition is equivalent to the discriminant being nonnegative: $$ 3^2 - 4b \geq 0. $$

Answer 2

$x^2 - 3x + b = (x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1r_2$ implies that $3 = r_1+r_2$ and $b = r_1r_2$