To complete the square for the polynomial \( k^2 + 10k \), we can follow these steps:
-
Start with the term \( k^2 + 10k \).
-
Take the coefficient of \( k \), which is \( 10 \), divide it by \( 2 \), and square the result:
\[ \left(\frac{10}{2}\right)^2 = 5^2 = 25 \]
-
Add and subtract this square inside the expression:
\[ k^2 + 10k + 25 - 25 \]
This rearranges to:
\[ (k + 5)^2 - 25 \]
Thus, the number that makes the polynomial a perfect-square quadratic is \( 25 \).
So the completed expression is \( k^2 + 10k + 25 \).