Question

Zavier, Boaz, and Sophie are trying to find the values of x that make the following quadratic function equal 0: x(x+2)=0
𝑥
(
𝑥
+
2
)
=
0
. Each student came up with a different answer, which they explained in the options. Which student’s explanation is correct?

Option #1: Zavier says that only x=−2
𝑥
=
−
2
will make the function equal 0, and he provided the given work: −2⋅(−2+2)=−2⋅(0)=0
−
2
⋅
(
−
2
+
2
)
=
−
2
⋅
(
0
)
=
0
.

Option #2: Boaz says that the values x=0
𝑥
=
0
and x=−2
𝑥
=
−
2
will both make the function equal 0, and he provided the given work: 0⋅(0+2)=0⋅(2)=0
0
⋅
(
0
+
2
)
=
0
⋅
(
2
)
=
0
, and −2⋅(−2+2)=−2(0)=0
−
2
⋅
(
−
2
+
2
)
=
−
2
(
0
)
=
0
.

Option #3: Sophie says that only x=0
𝑥
=
0
will make the function equal 0, and she provided the given work: 0⋅(0+2)=0⋅(2)=0
0
⋅
(
0
+
2
)
=
0
⋅
(
2
)
=
0
.

(1 point)
Option # is the correct explanation.

Answer 1

To solve the equation \( x(x+2) = 0 \), we can use the Zero Product Property, which states that if a product of two factors equals zero, then at least one of the factors must be zero.

The two factors in our equation are \( x \) and \( (x + 2) \). Setting each factor equal to zero gives us:

1. \( x = 0 \)
2. \( x + 2 = 0 \) which simplifies to \( x = -2 \)

Thus, the solutions to the equation are \( x = 0 \) and \( x = -2 \).

Now, let's evaluate each student's explanation:

- **Option #1 (Zavier)**: Claims that only \( x = -2 \) will make the function equal to 0. This is incorrect since there is also the solution \( x = 0 \).

- **Option #2 (Boaz)**: Correctly identifies both solutions as \( x = 0 \) and \( x = -2 \) and demonstrates work for both, which is accurate.

- **Option #3 (Sophie)**: Claims that only \( x = 0 \) will make the function equal to 0. This is also incorrect, as there is another solution \( x = -2 \).

Therefore, the correct explanation is **Option #2: Boaz**.