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.



Answer :

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].