Question

Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (5x + 1) (x - 8) = 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. In the case of the quadratic equation \((5x + 1)(x - 8) = 0\), we can apply this property.

To determine the statements that must be true based on the Zero Product Property, we need to set each factor equal to zero:

1. \(5x + 1 = 0\)
2. \(x - 8 = 0\)

Now, solve each equation:

1. For \(5x + 1 = 0\):
\[
5x = -1 \quad \Rightarrow \quad x = -\frac{1}{5}
\]

2. For \(x - 8 = 0\):
\[
x = 8
\]

Thus, the solutions to the quadratic equation \((5x + 1)(x - 8) = 0\) are \(x = -\frac{1}{5}\) and \(x = 8\).

Based on this, the statements that must be true are:

- The values \(x = -\frac{1}{5}\) and \(x = 8\) are the roots of the equation.
- At least one of the factors must equal zero.

If any other specific statements were included in your list, please provide those so I can verify which ones are true based on the context of the Zero Product Property.