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




(1 point)
Responses

No, there should be a solution #3.
No, there should be a solution #3.

No, x−9=0
should have been used to find solution #2.
No, x minus 9 is equal to 0 should have been used to find solution #2.

Yes, the student answered the problem correctly.
Yes, the student answered the problem correctly.

No, x+2=0
should have been used to find solution #1.
No, x plus 2 is equal to 0 should have been used to find solution #1.

Answer 1

The student’s work is not correct. Here’s what went wrong:

In the solution #2 step, the student incorrectly set up the equation to find the second solution. Instead of solving \( x + 9 = 0 \), the student should have used \( x - 9 = 0 \).

The correct steps should be:

1. From \( (x - 2)(x - 9) = 0 \), set each factor equal to zero:
   • \( x - 2 = 0 \) leads to \( x = 2 \) (this solution is correct)
   • \( x - 9 = 0 \) leads to \( x = 9 \) (this solution was missed)

Therefore, the correct conclusion is: No, there should be a solution #3.