To rewrite the quadratic function [tex]\( f(x) = x^2 + 10x + 37 \)[/tex] from standard form to vertex form, we will complete the square. 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.
Here are the steps to complete the square:
1. Start with the standard form of the quadratic function:
[tex]\[
f(x) = x^2 + 10x + 37
\][/tex]
2. Isolate the quadratic and linear terms:
[tex]\[
f(x) = (x^2 + 10x) + 37
\][/tex]
3. Add and subtract the square of half the coefficient of [tex]\( x \)[/tex] inside the parentheses:
The coefficient of [tex]\( x \)[/tex] is 10. Half of 10 is 5, and the square of 5 is [tex]\( 5^2 = 25 \)[/tex].
Adding and subtracting 25 inside the parentheses gives us:
[tex]\[
f(x) = (x^2 + 10x + 25 - 25) + 37
\][/tex]
4. Rewrite the expression inside the parentheses as a perfect square:
[tex]\[
f(x) = ((x + 5)^2 - 25) + 37
\][/tex]
5. Simplify the constant terms:
Combine [tex]\(-25\)[/tex] and [tex]\(37\)[/tex]:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
6. Write the function in vertex form:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
Hence, the quadratic function [tex]\( f(x) = x^2 + 10x + 37 \)[/tex] in vertex form is:
[tex]\[
f(x) = (x + 5)^2 + 12
\][/tex]
The vertex of the parabola described by this function is [tex]\((-5, 12)\)[/tex].
