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 find integers that make the equation \( (n - k + 2)^2 \) equal to the right side, we first simplify the left-hand side:

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

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

Now, expanding \( 2(n - k)(2) \):

\[
2(n - k)(2) = 4(n - k) = 4n - 4k
\]

Finally, we add \( 2^2 = 4 \) to the expression:

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

Now, we can rewrite it as:

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

Now, we can compare coefficients with the form:

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

By comparing, we find:

- Coefficient of \( n^2 \): **1**
- Coefficient of \( k^2 \): **1**
- Coefficient of \( n \): **4**
- Coefficient of \( k \): **-4**
- Coefficient of \( nk \): **-2**
- Constant term: **4**

Putting it all together:

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

Thus, the completed integers are:

1, 1, 4, -4, -2, 4.