Sure, let's rewrite the given function by completing the square.
1. Starting Function:
[tex]\[
f(x) = x^2 + 14x + 8
\][/tex]
2. Grouping the [tex]$x$[/tex] Terms:
To complete the square, we first look at the quadratic and linear terms: [tex]\(x^2 + 14x\)[/tex].
3. Forming a Perfect Square:
For the square term [tex]\((x + d)^2\)[/tex] to match [tex]\(x^2 + 14x\)[/tex], the constant term [tex]\(d\)[/tex] can be found using:
[tex]\[
d = \frac{b}{2a} = \frac{14}{2 \cdot 1} = 7
\][/tex]
Therefore, we want:
[tex]\[
(x + 7)^2 = x^2 + 14x + 49
\][/tex]
4. Adjusting the Constant Term:
Note that [tex]\((x + 7)^2\)[/tex] introduces an extra term [tex]\(49\)[/tex] that was not in the original equation, so we need to subtract this extra number:
[tex]\[
x^2 + 14x + 8 = (x + 7)^2 - 49 + 8
\][/tex]
5. Simplifying the Added/Subtracted Terms:
Now combine the constant terms:
[tex]\[
-49 + 8 = -41
\][/tex]
6. Final Function:
Therefore, we have:
[tex]\[
f(x) = (x + 7)^2 - 41
\][/tex]
So, the completed square form of the function is:
[tex]\[
f(x) = (x + 7)^2 - 41
\][/tex]
