To determine which expression [tex]$(m-3)^2$[/tex] is equivalent to, let's expand the expression step-by-step.
Given:
[tex]\[(m-3)^2\][/tex]
We can use the formula for the square of a binomial:
[tex]\[(a - b)^2 = a^2 - 2ab + b^2\][/tex]
Here, [tex]\(a = m\)[/tex] and [tex]\(b = 3\)[/tex]. So the expression becomes:
[tex]\[(m - 3)^2 = m^2 - 2 \cdot m \cdot 3 + 3^2\][/tex]
Let's solve it step-by-step:
1. First, we square the first term: [tex]\(m^2\)[/tex].
2. Next, we calculate the middle term: [tex]\(-2 \cdot m \cdot 3\)[/tex]. This simplifies to [tex]\(-6m\)[/tex].
3. Finally, we square the last term: [tex]\(3^2\)[/tex] which equals [tex]\(9\)[/tex].
Combining all these terms, we get:
[tex]\[m^2 - 6m + 9\][/tex]
Now, we compare this result with the given choices:
1. [tex]\(m^2 + 9\)[/tex] is not correct because it lacks the [tex]\(-6m\)[/tex] term.
3. [tex]\(m^2 - 6m + 9\)[/tex] matches our expanded expression perfectly.
2. [tex]\(m^2 - 9\)[/tex] is not correct because it is missing the [tex]\(6m\)[/tex] term and has a [tex]\(-9\)[/tex] instead of [tex]\(+9\)[/tex].
4. [tex]\(m^2 - 6m - 9\)[/tex] is not correct because it has [tex]\(-9\)[/tex] instead of [tex]\(+9\)[/tex].
Therefore, the correct choice is:
[tex]\[3) \ m^2 - 6m + 9\][/tex]
