To rewrite the quadratic function [tex]\( f(x) = x^2 + 4x + 41 \)[/tex] by completing the square, we need to transform it into the form [tex]\( f(x) = (x + a)^2 + b \)[/tex]. Here are the steps:
1. Identify and group the [tex]\( x \)[/tex]-terms:
[tex]\[
f(x) = x^2 + 4x + 41
\][/tex]
2. To complete the square, we first take the coefficient of [tex]\( x \)[/tex] (which is 4) and halve it:
[tex]\[
\frac{4}{2} = 2
\][/tex]
3. Next, we square this value:
[tex]\[
2^2 = 4
\][/tex]
4. Rewrite the function by adding and subtracting this square value inside the equation, ensuring that the function remains unchanged:
[tex]\[
f(x) = x^2 + 4x + 4 - 4 + 41
\][/tex]
5. Group the perfect square trinomial and calculate the remaining constant term:
[tex]\[
f(x) = (x + 2)^2 - 4 + 41
\][/tex]
6. Simplify the constant term:
[tex]\[
-4 + 41 = 37
\][/tex]
7. So the completed square form of the quadratic function is:
[tex]\[
f(x) = (x + 2)^2 + 37
\][/tex]
Thus, the quadratic function [tex]\( f(x) = x^2 + 4x + 41 \)[/tex] rewritten by completing the square is:
[tex]\[
f(x) = (x + 2)^2 + 37
\][/tex]
