Question

In the Zero Product Property, at least one of the factors must equal to zero or both? For example, if I have something like this:
(2x-3) (3x+6)=0
2x-3=0 and 3x+6=0

Answer 1

In the Zero Product Property, if the product of two or more factors equals zero, then at least one of the factors must equal zero. It is not necessary for both factors to equal zero; it is sufficient for only one of them to be zero for the entire product to be zero.

In your example, \((2x-3)(3x+6) = 0\), you can indeed set each factor equal to zero separately:

1. \(2x - 3 = 0\)
2. \(3x + 6 = 0\)

Solving these equations will give you the values of \(x\) that satisfy the original equation. So, you will solve:

1. \(2x - 3 = 0 \Rightarrow 2x = 3 \Rightarrow x = \frac{3}{2}\)
2. \(3x + 6 = 0 \Rightarrow 3x = -6 \Rightarrow x = -2\)

Thus, the solutions to the equation \((2x - 3)(3x + 6) = 0\) are \(x = \frac{3}{2}\) and \(x = -2\).