While using the Zero Product Property to find the solutions for the quadratic equation (x - 2) (x - 9) = 0, a student completed the work below. Is the student's work correct?

Question

(x - 2) (x - 9) = 0 original equation
Solution #1:
x - 2 = 0
x = 2
Solution #2:
x + 9 = 0
x = -9

Answer 1

The student's work contains an error in Solution #2. Let's go through the process correctly:

The original equation is: \[ (x - 2)(x - 9) = 0 \]

According to the Zero Product Property, if the product of two factors equals zero, at least one of the factors must be zero. Therefore, we need to set each factor equal to zero:

1. For the first factor: \[ x - 2 = 0 \] Adding 2 to both sides gives: \[ x = 2 \]

2. For the second factor: \[ x - 9 = 0 \quad \text{(not \( x + 9 = 0\))} \] Adding 9 to both sides gives: \[ x = 9 \]

So, the correct solutions to the equation \( (x - 2)(x - 9) = 0 \) are: \[ x = 2 \quad \text{and} \quad x = 9 \]

In conclusion, Solution #1 is correct, but Solution #2 is incorrect because the student incorrectly wrote \( x + 9 = 0 \) instead of \( x - 9 = 0 \). The correct solution from the second factor is \( x = 9 \).