To rewrite the quadratic function [tex]\( f(x) = x^2 + 10x + 37 \)[/tex] from standard form to vertex form, we follow the process of completing the square.
### Steps to Complete the Square:
1. Start with the standard form of the quadratic function:
[tex]\[
f(x) = x^2 + 10x + 37
\][/tex]
2. Group the [tex]\( x \)[/tex] terms together:
[tex]\[
f(x) = (x^2 + 10x) + 37
\][/tex]
3. Identify the coefficient of the [tex]\( x \)[/tex] term (which is 10 in this case), and divide it by 2, then square the result:
[tex]\[
\left(\frac{10}{2}\right)^2 = 5^2 = 25
\][/tex]
4. Add and subtract this square within the parenthesis to complete the square:
[tex]\[
f(x) = (x^2 + 10x + 25 - 25) + 37
\][/tex]
5. Rewrite the trinomial as a perfect square and simplify:
[tex]\[
f(x) = ((x + 5)^2 - 25) + 37
\][/tex]
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
### Vertex Form:
The quadratic function in vertex form is:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
### Identifying the Vertex:
The vertex form of a quadratic function is [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h,k)\)[/tex] is the vertex of the parabola.
In our equation:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
The vertex [tex]\((h,k)\)[/tex] is:
[tex]\[
h = -5, \quad k = 12
\][/tex]
### Final Answer:
The quadratic function [tex]\( f(x) = x^2 + 10x + 37 \)[/tex] rewritten in vertex form is:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
And the vertex of this quadratic function is:
[tex]\[
(-5, 12)
\][/tex]
