To factorize the quadratic expression [tex]\( x^2 - 5x + 6 \)[/tex] using a geometric model involving algebra tiles, let's go through the steps in detail.
### Step 1: Set Up the Expression
We start by identifying the components of the quadratic expression:
- [tex]\( x^2 \)[/tex]: This term is represented by a large square tile.
- [tex]\(-5x\)[/tex]: This term consists of five unit-length rectangular tiles, each with a dimension of [tex]\( x \)[/tex] by [tex]\( 1 \)[/tex], but since it is negative, we can think of them as "negative" tiles.
- [tex]\( 6 \)[/tex]: This term is represented by six small unit square tiles.
### Step 2: Arrange the Tiles
Our goal is to arrange the tiles into a rectangular form since that will help us visualize the product of two binomials.
#### Step 2.1: Place the [tex]\( x^2 \)[/tex] Tile
Place the large square tile representing [tex]\( x^2 \)[/tex] first.
#### Step 2.2: Position the [tex]\( -5x \)[/tex] Tiles
Next, we need to place the five [tex]\( -x \)[/tex] tiles. We'll try to place these along the sides of the large square tile in such a way as to form an incomplete rectangle.
#### Step 2.3: Position the [tex]\( 6 \)[/tex] Unit Tiles
Finally, place the six unit tiles to complete the sides around the [tex]\( x^2 \)[/tex] tile, filling in gaps and trying to form a complete rectangle.
### Step 3: Visualize the Rectangle and Factors
The objective is to complete the rectangle so that it becomes apparent which binomials multiply to give the original quadratic expression. Here's how:
1. Arrange such that sides of [tex]\( x^2 \)[/tex]: Try finding two pairs of numbers that multiply to 6 (constant term) and add to -5 (the middle term).
- The pairs that work for this are [tex]\( -2 \)[/tex] and [tex]\( -3 \)[/tex].
2. Forming the Binomials:
- [tex]\( x - 2 \)[/tex]: Represents taking [tex]\( x \)[/tex] units along one side and subtracting 2.
- [tex]\( x - 3 \)[/tex]: Represents taking [tex]\( x \)[/tex] units along another side and subtracting 3.
Combining these, we get:
- Place the [tex]\( x^2 \)[/tex] tile at the corner.
- Along one side, place [tex]\( x - 2 \)[/tex] tiles.
- Along the other side, place [tex]\( x - 3 \)[/tex] tiles.
### Step 4: Verification
To confirm, check the area of the formed rectangle:
- The length and width of the rectangle are both expressed in terms of [tex]\( x - 2 \)[/tex] and [tex]\( x - 3 \)[/tex].
- The total area is [tex]\( (x - 2)(x - 3) = x^2 - 5x + 6 \)[/tex].
Therefore, the correct factorization of [tex]\( x^2 - 5x + 6 \)[/tex] is [tex]\( (x - 2)(x - 3) \)[/tex], and it can be represented visually as a rectangle using algebra tiles, with the sides being divided into [tex]\( x - 2 \)[/tex] and [tex]\( x - 3 \)[/tex].
### Conclusion
The algebra tiles forming a geometric model represent the factorization of [tex]\( x^2 - 5x + 6 \)[/tex] as [tex]\( (x - 2)(x - 3) \)[/tex].
