To write [tex]\( h(x) = x^2 - 6x + 3 \)[/tex] in vertex form, we will complete the square. Here is a detailed, step-by-step process to achieve this.
### Step 1: Setting Up the Expression
Given the quadratic expression:
[tex]\[ h(x) = x^2 - 6x + 3 \][/tex]
We can start by focusing on the quadratic and linear terms:
[tex]\[ x^2 - 6x \][/tex]
### Step 2: Completing the Square
1. Identify the coefficient of [tex]\( x \)[/tex]: The coefficient of [tex]\( x \)[/tex] is [tex]\(-6\)[/tex].
2. Take half of the coefficient of [tex]\( x \)[/tex]: Half of [tex]\(-6\)[/tex] is [tex]\(-3\)[/tex].
3. Square this value:
[tex]\[
(-3)^2 = 9
\][/tex]
4. Add and subtract this square within the expression: We adjust our expression to maintain equality:
[tex]\[
x^2 - 6x + 9 - 9 + 3
\][/tex]
In this step, we essentially add and subtract [tex]\( 9 \)[/tex] to ensure the equation remains balance.
5. Rewrite the expression: Now write it as a perfect square minus the outside value:
[tex]\[
x^2 - 6x + 9 - 9 + 3 = (x - 3)^2 - 9 + 3
\][/tex]
### Step 3: Simplify
Combine the constants:
[tex]\[
(x - 3)^2 - 9 + 3 = (x - 3)^2 - 6
\][/tex]
So, the quadratic expression [tex]\( x^2 - 6x + 3 \)[/tex] in vertex form is:
[tex]\[ h(x) = (x - 3)^2 - 6 \][/tex]
### Summary
The vertex form of the quadratic function [tex]\( h(x) = x^2 - 6x + 3 \)[/tex] is:
[tex]\[ h(x) = (x - 3)^2 - 6 \][/tex]
The vertex of this quadratic function is at the point [tex]\( (3, -6) \)[/tex].
