Two students in Ms. Chau's class were asked to write an equivalent expression for the expression (a−6)(a−4)

Question

. Identify the student with the correct work and where the mistakes were in the other student's work.

(1 point)
Responses

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.

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.

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.
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Answer 1

To solve for the equivalent expression for \((a−6)(a−4)\), we need to apply the distributive property (also known as the FOIL method for binomials):

\[ (a−6)(a−4) = a \cdot a + a \cdot (-4) + (-6) \cdot a + (-6) \cdot (-4) \]

Calculating each term:

1. \(a \cdot a = a^2\)
2. \(a \cdot (-4) = -4a\)
3. \((-6) \cdot a = -6a\)
4. \((-6) \cdot (-4) = 24\)

Combining these results, we get:

\[ a^2 - 4a - 6a + 24 = a^2 - 10a + 24 \]

Now, let's analyze the responses:

• Gabrielle is correct in the multiplication of constants; \((-6) \cdot (-4) = 24\).
• Brittany could have gotten \(-24\) if she made a mistake in that multiplication.

Given these statements:

1. Gabrielle is correct; Brittany made an error in multiplying the constants \(-6\) and \(-4\), possibly getting \(-24\) instead of \(+24\).
2. Brittany is incorrect; Gabrielle’s result of \(+24\) is accurate.

Thus, the correct response to identify the students' work is:

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