Bellshaffonidn
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### Completing the Square to Write in Vertex Form

Write [tex]\( h(x) = x^2 - 6x + 3 \)[/tex] in vertex form by following these steps:

Step 1: Model [tex]\( x^2 - 6x + 3 \)[/tex] by placing tiles in the Product section. First, place the [tex]\( x^2 \)[/tex] and [tex]\( -x \)[/tex] tiles along the left and top borders to form two sides of a square. Then place the [tex]\( + \)[/tex] tiles along the ends of the [tex]\( -x \)[/tex] tiles.

Step 2: Form a square of tiles in the Product section. Add as many zero-pair tiles as necessary. Move the extra [tex]\( - \)[/tex] tiles off to the side.

Sagot :

GinnyAnsweridn
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.
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