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How could Brent use a rectangle to model the factors of [tex]$x^2 - 7x + 6$[/tex]?

A. He could draw a diagram of a rectangle with dimensions [tex]$x-3$[/tex] and [tex][tex]$x-4$[/tex][/tex] and then show the area is equivalent to the sum of [tex]$x^2, -3x, -4x,$[/tex] and [tex]12$[/tex].

B. He could draw a diagram of a rectangle with dimensions [tex]$x+7[tex]$[/tex] and [tex]$[/tex]x-1$[/tex] and then show the area is equivalent to the sum of [tex]$x^2, 7x, -x,$[/tex] and [tex]6$[/tex].

C. He could draw a diagram of a rectangle with dimensions [tex][tex]$x-1$[/tex][/tex] and [tex]$x-6$[/tex] and then show the area is equivalent to the sum of [tex]$x^2, -x, -6x,$[/tex] and [tex]6$[/tex].

D. He could draw a diagram of a rectangle with dimensions [tex]$x-4[tex]$[/tex] and [tex]$[/tex]x+3$[/tex] and then show the area is equivalent to the sum of [tex]$x^2, -4x, 3x,$[/tex] and [tex]-12$[/tex].



Answer :

GinnyAnswer
To model the factors of [tex]\(x^2 - 7x + 6\)[/tex] using a rectangle, Brent should do the following:

1. Identify potential factors of the quadratic expression [tex]\(x^2 - 7x + 6\)[/tex]. The expression can be factored into [tex]\((x - 1)(x - 6)\)[/tex].

2. Draw a rectangle with dimensions corresponding to these factors. So, he should draw a rectangle with one side labeled [tex]\(x - 1\)[/tex] and the other side labeled [tex]\(x - 6\)[/tex].

3. Calculate the area of the rectangle, which can be represented as the product of [tex]\(x - 1\)[/tex] and [tex]\(x - 6\)[/tex].

4. Expand this product to verify it corresponds to the original quadratic expression.

Here's the step-by-step process:

### Step 1: Draw the Rectangle

Brent should draw a rectangle with dimensions [tex]\(x - 1\)[/tex] and [tex]\(x - 6\)[/tex].

### Step 2: Set Up the Area Calculation

To visualize how the product [tex]\((x - 1)(x - 6)\)[/tex] matches the original quadratic expression, we can split the calculation into parts:

[tex]\[ (x - 1)(x - 6) \][/tex]

### Step 3: Expand the Product

Use the distributive property (also known as the FOIL method for binomials) to expand the expression:

[tex]\[ (x - 1)(x - 6) = x \cdot x + x \cdot (-6) + (-1) \cdot x + (-1) \cdot (-6) \][/tex]

Simplify each term:

[tex]\[ x^2 - 6x - x + 6 \][/tex]

### Step 4: Combine Like Terms

Combine the middle terms [tex]\(-6x\)[/tex] and [tex]\(-x\)[/tex]:

[tex]\[ x^2 - 7x + 6 \][/tex]

### Conclusion

By verifying the expanded product, we can see that the area of the rectangle with dimensions [tex]\(x - 1\)[/tex] and [tex]\(x - 6\)[/tex] matches the given quadratic expression [tex]\(x^2 - 7x + 6\)[/tex]. Thus, Brent should draw a diagram of a rectangle with dimensions [tex]\(x - 1\)[/tex] and [tex]\(x - 6\)[/tex] and show the area is equivalent to the sum of [tex]\(x^2, -x, -6x\)[/tex], and [tex]\(6\)[/tex].

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