To complete the square for the given polynomial [tex]\( x^2 + 11x \)[/tex], we follow these steps:
1. Identify the coefficient of the linear term: The coefficient of [tex]\( x \)[/tex] in the polynomial [tex]\( x^2 + 11x \)[/tex] is 11.
2. Divide the coefficient by 2:
[tex]\[
\frac{11}{2} = 5.5
\][/tex]
3. Square the result:
[tex]\[
(5.5)^2 = 30.25
\][/tex]
4. Add and subtract this squared term inside the polynomial to complete the square:
[tex]\[
x^2 + 11x + 30.25 - 30.25
\][/tex]
5. Rewrite the expression by grouping the complete square and the constant term:
[tex]\[
x^2 + 11x + 30.25 = (x + 5.5)^2
\][/tex]
So, to complete the square, we add [tex]\( 30.25 \)[/tex] to the given polynomial. This transforms the polynomial into the form:
[tex]\[
x^2 + 11x + 30.25 = (x + 5.5)^2
\][/tex]
Thus, the missing term is [tex]\( 30.25 \)[/tex], and the factored form is [tex]\( (x + 5.5)^2 \)[/tex].
So the completed expression is:
[tex]\[
x^2 + 11x + 30.25 = (x + 5.5)^2
\][/tex]
