Final answer:
To factor the quadratic equation x^2 - 3x - 4, we find numbers that multiply to -4 (ac) and add to -3 (b), which are -4 and +1. We then rewrite the middle term, group common factors, and factor out the common binomial, resulting in (x - 4)(x + 1).
Step-by-step explanation:
To factor the quadratic equation x^2 - 3x - 4 using the AC Method, you need to express this equation in the form ax^2 + bx + c = 0, where a, b, and c are integers, and then find two numbers that multiply to ac (the coefficient of x^2 times the constant term) and add up to b (the coefficient of x). The equation x^2 - 3x - 4 has a = 1, b = -3, and c = -4, so ac = -4 and we are looking for two numbers that multiply to -4 and add to -3.
The two numbers that meet these criteria are -4 and +1 because (-4) * (+1) = -4 and (-4) + (+1) = -3. Now we can rewrite the middle term (bx) using these two numbers: x^2 - 4x + x - 4. Next we group the terms: (x^2 - 4x) + (x - 4), and factor by grouping.
The first group has a common factor of x: x(x - 4), and the second group's common factor is +1: +1(x - 4). Now, both groups have a common binomial factor of (x - 4), so we factor this out, resulting in (x - 4)(x + 1).
Hence, the factored form of the equation x^2 - 3x - 4 is (x - 4)(x + 1), which can also be checked by foiling the binomials to see that they equal the original equation.