Question

Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (3x + 4) (x − 2) = 0
A) No values of x will make the function equal to 0.
B) Neither of the factors, (3x + 4) or (x - 2), must equal 0 for the entire function to equal 0.
C) Both factors, (3x + 4) and (x - 2), must equal 0 for the entire function to equal 0.
D) At least one of the factors (3x + 4) or (x - 2), must equal 0 for the entire function to equal 0.

Answer 1

The correct answer based on the Zero Product Property is:

**D) At least one of the factors (3x + 4) or (x - 2), must equal 0 for the entire function to equal 0.**

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. In this case, for the equation (3x + 4)(x - 2) = 0 to hold true, either:

1. \(3x + 4 = 0\) or
2. \(x - 2 = 0\)

must be true. Therefore, at least one of these factors must equal zero for the equation to equal zero.

The other options are not correct:
- A) is false because there are values of x that will make the function equal to 0.
- B) is false because at least one factor must equal 0 for the product to equal 0.
- C) is false because only one factor needs to equal 0, not both.