Question

Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (3x + 4) (x - 2) = 0

Answer 1

The Zero Product Property states that if a product of two factors equals zero, then at least one of the factors must be equal to zero. In the case of the quadratic equation \((3x + 4)(x - 2) = 0\), we can apply the Zero Product Property as follows:

1. Set each factor equal to zero:
- \(3x + 4 = 0\)
- \(x - 2 = 0\)

2. Solve each equation:
- For \(3x + 4 = 0\):
\[
3x = -4 \implies x = -\frac{4}{3}
\]
- For \(x - 2 = 0\):
\[
x = 2
\]

Therefore, the solutions to the quadratic equation are \(x = -\frac{4}{3}\) and \(x = 2\).

Based on the Zero Product Property, we can conclude that at least one of the factors must be zero for the product to equal zero. Thus, the following statements must be true:
- Either \(3x + 4 = 0\) (yielding \(x = -\frac{4}{3}\)) or \(x - 2 = 0\) (yielding \(x = 2\)).

In conclusion, the true statements about the quadratic equation (3x + 4)(x - 2) = 0 are:
- Either \(x = -\frac{4}{3}\) or \(x = 2\).