IDNLearn.com makes it easy to get reliable answers from experts and enthusiasts alike. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
Sagot :
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].
### 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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
