To write the function [tex]\( h(x) = x^2 - 6x + 3 \)[/tex] in vertex form, we use the method of completing the square. Here's a detailed, step-by-step approach:
### Step 1: Understand the Initial Function
We start with the quadratic function:
[tex]\[ h(x) = x^2 - 6x + 3 \][/tex]
### Step 2: Arrange the Terms for Completing the Square
To complete the square, we look at the quadratic and linear terms, [tex]\( x^2 - 6x \)[/tex].
### Step 3: Halve the Coefficient of [tex]\( x \)[/tex] and Square It
Take half the coefficient of [tex]\( x \)[/tex], which is [tex]\(-6\)[/tex]. Halving it gives [tex]\(-3\)[/tex]. Squaring [tex]\(-3\)[/tex] gives:
[tex]\[ (-3)^2 = 9 \][/tex]
### Step 4: Add and Subtract This Square
Add and subtract this square inside the function:
[tex]\[ h(x) = x^2 - 6x + 9 - 9 + 3 \][/tex]
### Step 5: Group to Form a Perfect Square
Combine the first three terms to form a perfect square:
[tex]\[ h(x) = (x - 3)^2 - 9 + 3 \][/tex]
### Step 6: Simplify
Simplify the constant terms:
[tex]\[ h(x) = (x - 3)^2 - 6 \][/tex]
### Final Form in Vertex Form
The vertex form of [tex]\( h(x) = x^2 - 6x + 3 \)[/tex] is thus:
[tex]\[ h(x) = (x - 3)^2 - 6 \][/tex]
### Verification
To verify, you can expand the vertex form to see if it matches the original function:
[tex]\[ (x - 3)^2 - 6 = x^2 - 6x + 9 - 6 = x^2 - 6x + 3 \][/tex]
Thus, the vertex form of the given quadratic function is:
[tex]\[ h(x) = (x - 3)^2 - 6 \][/tex]
This shows the correct transformation using the completing the square method.
