Fill in the blanks to get an identity. (Each blank should contain an integer.)



(n - k + 2)^2 = __ n^2 + ___ k^2 + ___ n + ___k + ___ nk + ___

1 answer

To determine the integers that fit into the blanks and make the given identity true, we need to expand the left-hand side of the equation \((n - k + 2)^2\) and match it to the right-hand side in the desired form \(a n^2 + b k^2 + c n + d k + e nk + f\).

Start by expanding \((n - k + 2)^2\):

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

Apply the distributive property:

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

Distribute each term inside the parentheses:

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

Combine like terms:

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

Now, compare this result with the desired form \(a n^2 + b k^2 + c n + d k + e nk + f\):

\[
n^2 + k^2 - 2nk + 4n - 4k + 4
\]

We can see that:
- \(a = 1\) (coefficient of \(n^2\))
- \(b = 1\) (coefficient of \(k^2\))
- \(c = 4\) (coefficient of \(n\))
- \(d = -4\) (coefficient of \(k\))
- \(e = -2\) (coefficient of \(nk\))
- \(f = 4\) (constant term)

Thus, the completed identity is:

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

So, filling in the blanks, we get:

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