How could Brent use a rectangle to model the factors of x2 – 7x + 6

How could Brent use a rectangle to model the factors of x2 – 7x + 6?

The correct answer and explanation is:

To model the factors of the quadratic expression x² – 7x + 6 using a rectangle, Brent can use the area model (also called the box method) which visually represents the process of factoring trinomials. Here’s how:

✅ Correct Answer:

Brent can draw a rectangle and divide it into four sections. He places x² in one box (representing the area), 6 in another (the constant term), and then finds two numbers that multiply to 6 (the constant) and add up to –7 (the coefficient of the middle term). These two numbers are –6 and –1. He then writes –6x and –1x in the remaining boxes. Finally, he factors out the greatest common factor from each row and column to get (x – 6)(x – 1).

🧠 Explanation (300 words):

Factoring a quadratic expression like x² – 7x + 6 means finding two binomials whose product gives the original expression. Brent can use a rectangle (box) to represent the terms of the quadratic and organize them to identify common factors visually.

1. Step 1: Draw a rectangle and divide it into four boxes.
   The expression has three terms: x² (leading term), –7x (middle term), and 6 (constant). The product of the first and last terms is x² × 6 = 6x².
2. Step 2: Find two numbers that multiply to 6 (constant term) and add up to –7 (coefficient of the middle term).
   The numbers –6 and –1 satisfy this because:
   • (–6) × (–1) = 6
   • (–6) + (–1) = –7
3. Step 3: Place x², –6x, –1x, and 6 in the four boxes of the rectangle.
   • Top left: x²
   • Top right: –6x
   • Bottom left: –1x
   • Bottom right: 6
4. Step 4: Factor each row and column.
   • The common factors from rows and columns give the binomials:
     (x – 6)(x – 1)

This rectangle model helps Brent see how the expression splits into parts that can be grouped and factored easily. It’s especially useful for visual learners and helps build foundational algebra skills.