Question

Which constant term would mean that the expression is completely factored?
x2 – 3x +

Answer 1

Do you mean factorable into a perfect square, or just factorable over the integers? Both answers are useful:

1) As a perfect square: complete square of the form (x − a)^2 = x^2 − 2ax + a^2.
Here −2a = −3 → a = 3/2, so the constant must be 9/4.
x^2 − 3x + 9/4 = (x − 3/2)^2.

2) Factorable over the integers: the discriminant 9 − 4c must be a perfect square. Let 9 − 4c = k^2 (k integer). Solving gives c = (9 − k^2)/4. Integer choices include:
- k = 3 → c = 0, x^2 − 3x = x(x − 3)
- k = 1 → c = 2, x^2 − 3x + 2 = (x − 1)(x − 2)
- k = 5 → c = −4, x^2 − 3x − 4 = (x − 4)(x + 1)

If you meant something else, tell me which domain (integers, rationals, reals) you want factors in.