To rewrite the function [tex]\( f(x) = x^2 + 20x + 40 \)[/tex] by completing the square, follow these steps:
1. Identify the coefficient of [tex]\(x\)[/tex]: For the quadratic term [tex]\(x^2\)[/tex], the coefficient of [tex]\(x\)[/tex] is 20.
2. Take half the coefficient of [tex]\(x\)[/tex]: Half of 20 is 10.
3. Square this value: The square of 10 is 100.
4. Rewrite the middle term: Add and subtract this square (100) within the function to complete the square.
[tex]\[
f(x) = x^2 + 20x + 40
\][/tex]
Add and subtract 100:
[tex]\[
f(x) = x^2 + 20x + 100 - 100 + 40
\][/tex]
5. Form the perfect square trinomial:
[tex]\[
f(x) = (x + 10)^2 - 60
\][/tex]
So the function [tex]\( f(x) \)[/tex] rewritten by completing the square is:
[tex]\[
f(x) = (x + 10)^2 - 60
\][/tex]
