Question

Two students in Ms. Chau's class were asked to write an equivalent expression for the expression (a−6)(a−4)
. Identify the student with the correct work and where the mistakes were in the other student's work.
Gabrielle's work:
(a - 6)(a - 4)
= a x a + a x (-4) - 6 x (a) - 6 x (-4)
= a^2 - 4a - 6a - 24
= a^2 - 10a - 24

Brittany's work:
(a - 6)(a - 4)
= a x a - 4a - 6a - 6 x (-4)
= a^2 - 10a + 24
Brittany is correct; Gabrielle used arrows to show the distributive property.
Brittany is correct; Gabrielle used arrows to show the distributive property.

Gabrielle is correct; Brittany multiplied the constants −6⋅−4
and got +24
, and she should have gotten −24
.
Gabrielle is correct; Brittany multiplied the constants negative 6 times negative 4 and got plus 24, and she should have gotten negative 24.

Brittany is correct; Gabrielle multiplied the constants −6⋅−4
and got −24
, and she should have gotten +24
.
Brittany is correct; Gabrielle multiplied the constants negative 6 times negative 4 and got negative 24, and she should have gotten plus 24.

Gabrielle is correct; Brittany did not use arrows to show the distributive property before she multiplied.
Gabrielle is correct; Brittany did not use arrows to show the distributive property before she multiplied.

Answer 1

Let's analyze both students' work for any mistakes and clarify who is correct.

**Gabrielle's work:**
1. She starts with \((a - 6)(a - 4)\).
2. Applies the distributive property (though not using arrows).
3. Writes:
\[
= a \times a + a \times (-4) - 6 \times (a) - 6 \times (-4)
\]
which can be simplified to:
\[
= a^2 - 4a - 6a + 24
\]
(Note: Gabrielle wrote \(- 24\) instead of \(+ 24\) because she miscalculated the product of \(-6\) and \(-4\) in the last term.)
4. Finally, she sums up to:
\[
= a^2 - 10a - 24
\]

**Brittany's work:**
1. She also starts with \((a - 6)(a - 4)\).
2. She distributes correctly with the expression:
\[
= a \times a - 4a - 6a + 24
\]
(as she correctly calculates \(-6 \times -4 = +24\)).
3. She ends up with:
\[
= a^2 - 10a + 24
\]

**Conclusion:**
1. Gabrielle made an error in the final constant term by concluding that \(-6 \times -4 = -24\) instead of \(24\).
2. Brittany’s calculated expression \(a^2 - 10a + 24\) is indeed correct.

**Correct Identification:**
Brittany is correct; Gabrielle calculated the constants \(-6 \cdot -4\) incorrectly and got \(-24\) when it should have been \(+24\).

So, the correct statement is:
**Brittany is correct; Gabrielle multiplied the constants \(-6 \cdot -4\) and got \(-24\), and she should have gotten \(+24\).**