Question

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

Answer 1

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must equal zero. For the quadratic equation (2x - 3)(x + 2) = 0, we can apply this property.

According to the Zero Product Property, we can set each factor equal to zero:

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

Now we can solve these individual equations:

1. From \( 2x - 3 = 0 \):
\[
2x = 3 \\
x = \frac{3}{2}
\]

2. From \( x + 2 = 0 \):
\[
x = -2
\]

Thus, the statements that must be true are:

- The equation has solutions \( x = \frac{3}{2} \) and \( x = -2 \).
- These two values are the x-intercepts of the quadratic function represented by the equation.

In summary, the statement that must be true is that the solutions to the equation are \( x = \frac{3}{2} \) and \( x = -2 \).