Question

Fill in the blanks with integers to get an identity.
(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k + ___ nk + ___

Answer 1

To fill in the blanks for the identity \((n - k + 2)^2\), we first need to expand the left-hand side.

Starting with the expression:

\[
(n - k + 2)^2
\]

We can expand it using the formula for the square of a binomial \((a + b + c)^2\):

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\]

Here, we let:
- \(a = n\)
- \(b = -k\)
- \(c = 2\)

Now expanding this gives us:

\[
(n - k + 2)^2 = n^2 + (-k)^2 + 2^2 + 2(n)(-k) + 2(n)(2) + 2(-k)(2)
\]

Calculating each term:

- \(n^2\) remains as \(n^2\)
- \((-k)^2 = k^2\)
- \(2^2 = 4\)
- \(2(n)(-k) = -2nk\)
- \(2(n)(2) = 4n\)
- \(2(-k)(2) = -4k\)

So when we combine all these terms, we have:

\[
(n - k + 2)^2 = n^2 + k^2 + 4 + (-2nk) + (4n) + (-4k)
\]

Rearranging this, we get:

\[
(n - k + 2)^2 = n^2 + k^2 + 4 + 4n - 4k - 2nk
\]

Now we can fill in the blanks:
1. The coefficient of \(n^2\) is \(1\).
2. The coefficient of \(k^2\) is \(1\).
3. The coefficient of \(n\) is \(4\).
4. The coefficient of \(k\) is \(-4\).
5. The coefficient of \(nk\) is \(-2\).
6. The constant term is \(4\).

So, we get:

\[
(n - k + 2)^2 = 1 \cdot n^2 + 1 \cdot k^2 + 4 \cdot n - 4 \cdot k - 2 \cdot nk + 4
\]

Thus the filled blanks are:

\[
1, 1, 4, -4, -2, 4
\]

So, the final answer is:

\[
(n - k + 2)^2 = 1n^2 + 1k^2 + 4n - 4k - 2nk + 4
\]