Rectangle Area Comparison: Finding Difference Between (x+4)(x+3) and (x+5)(x+2)

Q: Why can't I just multiply the numbers and ignore x?

Because x represents a variable length! The rectangles' areas depend on x. You must expand the entire expression $ (x+4)(x+3) $ to get the complete area formula.

Q: What's the easiest way to expand these expressions?

Use FOIL method: First terms (x·x), Outer terms (x·3), Inner terms (4·x), Last terms (4·3). Then combine like terms!

Q: How do I know which rectangle is bigger without knowing x?

Compare the expanded forms! Rectangle A gives $ x^2 + 7x + 12 $ and Rectangle B gives $ x^2 + 7x + 10 $. Since 12 > 10, Rectangle A is always 2 units larger.

Q: Why does the x² and 7x cancel out when comparing?

Because both rectangles have identical x² and 7x terms! When you subtract one area from the other, these identical parts cancel out, leaving only the constant difference of 12 - 10 = 2.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine which area is larger

00:03 We'll use the formula for calculating the area of a rectangle (length times width)

00:08 We'll substitute appropriate values according to the given data and calculate the area

00:13 Open parentheses properly, multiply each factor by each factor

00:23 Combine the terms

00:30 This is the expression for rectangle 1's area, now let's move to the second one

00:34 Again we'll use the formula and multiply the sides

00:38 Open parentheses properly, multiply each factor by each factor

00:49 Combine the terms

00:55 Compare the terms of the expressions, and find which is larger

01:00 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the rectangles in the diagram below.

Which has a larger area and by how much?

2

Step-by-step solution

Let's calculate the area of each rectangle step by step:

Step 1: Calculate the area of Rectangle A.
- Dimensions are $(x+4)$ and $(x+3)$ .
- The area is calculated as $(x+4) \times (x+3)$ .

Expanding this expression using the distributive property, we get:
$(x+4)(x+3) = x \cdot x + x \cdot 3 + 4 \cdot x + 4 \cdot 3$
$\Rightarrow x^2 + 3x + 4x + 12$
$\Rightarrow x^2 + 7x + 12$

Step 2: Calculate the area of Rectangle B.
- Dimensions are $(x+5)$ and $(x+2)$ .
- The area is calculated as $(x+5) \times (x+2)$ .

Using the distributive property to expand:
$(x+5)(x+2) = x \cdot x + x \cdot 2 + 5 \cdot x + 5 \cdot 2$
$\Rightarrow x^2 + 2x + 5x + 10$
$\Rightarrow x^2 + 7x + 10$

Step 3: Compare the areas of Rectangle A and B.
- Area of Rectangle A: $x^2 + 7x + 12$
- Area of Rectangle B: $x^2 + 7x + 10$

Subtract the area of Rectangle B from the area of Rectangle A:
$(x^2 + 7x + 12) - (x^2 + 7x + 10) = (x^2 + 7x + 12) - x^2 - 7x - 10$
$\Rightarrow x^2 - x^2 + 7x - 7x + 12 - 10$
$\Rightarrow 2$

Thus, the area of Rectangle A is larger by $2$ area units.

The correct answer, therefore, is: The area of rectangle A is larger by 2 area units.

3

Final Answer

The area of rectangle A is larger by 2 area units.

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle area equals length times width using FOIL method
Technique: Expand (x+4)(x+3) = x² + 7x + 12 systematically
Check: Subtract final expressions: (x² + 7x + 12) - (x² + 7x + 10) = 2 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute all terms when expanding
Don't just multiply x·x and 4·3 = x² + 12! This skips the middle terms and gives wrong areas. Always use FOIL: First, Outer, Inner, Last to get all four products like x·x + x·3 + 4·x + 4·3.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why can't I just multiply the numbers and ignore x?

+

Because x represents a variable length! The rectangles' areas depend on x. You must expand the entire expression $(x+4)(x+3)$ to get the complete area formula.

What's the easiest way to expand these expressions?

+

Use FOIL method: First terms (x·x), Outer terms (x·3), Inner terms (4·x), Last terms (4·3). Then combine like terms!

How do I know which rectangle is bigger without knowing x?

+

Compare the expanded forms! Rectangle A gives $x^2 + 7x + 12$ and Rectangle B gives $x^2 + 7x + 10$ . Since 12 > 10, Rectangle A is always 2 units larger.

Why does the x² and 7x cancel out when comparing?

+

Because both rectangles have identical x² and 7x terms! When you subtract one area from the other, these identical parts cancel out, leaving only the constant difference of 12 - 10 = 2.

Can I use specific numbers for x to check my answer?

+

Absolutely! Try x = 1: Rectangle A = 5×4 = 20, Rectangle B = 6×3 = 18. The difference is 20-18 = 2, which matches our algebraic result!

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Rectangle Area Comparison: Finding Difference Between (x+4)(x+3) and (x+5)(x+2)

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