Area Comparison: Square vs. Modified Rectangle with Side X cm

Q: Why can't I just compare X+3 and X-3 to determine which area is larger?

You need to compare the areas, not just the side lengths! Area involves multiplication of dimensions, so you must calculate $ X^2 $ versus $ (X+3)(X-3) $.

Q: What if X = 3? Would the rectangle still be smaller?

The problem states $ X > 3 $, so X cannot equal 3. If X were 3, one side would become 0 cm, making it impossible to form a rectangle!

Q: Does the difference of 9 cm² stay the same for any value of X?

Yes! No matter what value X has (as long as X > 3), the square will always have exactly 9 cm² more area than the rectangle.

Q: How do I remember the difference of squares formula?

Think of it as (First + Last)(First - Last) = First² - Last². The middle terms cancel out when you expand: $ (X+3)(X-3) = X^2 - 3X + 3X - 9 = X^2 - 9 $

Area Comparison with Algebraic Expressions

The side length of a square is X cm

$(x>3)$

We extend one side by 3 cm and shorten an adjacent side by 3 cm, and we get a rectangle.

Which shape has a larger area?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Which square has a larger area?

00:03 Let's draw the square, its sides are greater than 3

00:08 Let's draw the new rectangle according to the given data

00:24 We added and subtracted 3 from the appropriate sides according to the given data

00:31 We'll use the formula for calculating the area of a square (side squared)

00:34 We'll substitute appropriate values and solve to find the area

00:37 Let's calculate the rectangle's area (length times width)

00:40 We'll substitute appropriate values and solve to find the rectangle's area

00:47 Let's make sure to use parentheses properly

00:53 The rectangle's area equals the square's area minus 9, therefore it's smaller

00:57 And that's the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The side length of a square is X cm

$(x>3)$

We extend one side by 3 cm and shorten an adjacent side by 3 cm, and we get a rectangle.

Which shape has a larger area?

Step-by-step solution

To determine which shape has a larger area, we need to compare the areas of the square and the rectangle:

Step 1: Calculate the area of the original square:

The side length of the square is $X$ , so its area is given by:

\text{Area of square} = X^2

Step 2: Calculate the area of the rectangle:

The dimensions of the rectangle are $X + 3$ cm and $X - 3$ cm. Thus, its area is:

\text{Area of rectangle} = (X + 3)(X - 3)

Using the difference of squares formula, we find:

(X + 3)(X - 3) = X^2 - 9

Step 3: Compare the areas:

We compute the difference between the square's area and the rectangle's area:

X^2 - (X^2 - 9) = 9

Since 9 is positive, the area of the square is larger than the area of the rectangle.

Therefore, the square has a larger area than the rectangle.

Final Answer

The square

Key Points to Remember

Essential concepts to master this topic

Area Formula: Square area is $X^2$ , rectangle area is length × width
Technique: Use difference of squares: $(X+3)(X-3) = X^2 - 9$
Check: Compare areas by subtraction: $X^2 - (X^2-9) = 9 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Expanding (X+3)(X-3) incorrectly
Don't expand as X² + 6X - 9 = wrong rectangle area! This ignores the difference of squares pattern and creates extra terms. Always use the formula (a+b)(a-b) = a² - b² directly.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side AB is 2 cm long and side BC has a length of 7 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why can't I just compare X+3 and X-3 to determine which area is larger?

You need to compare the areas, not just the side lengths! Area involves multiplication of dimensions, so you must calculate $X^2$ versus $(X+3)(X-3)$ .

What if X = 3? Would the rectangle still be smaller?

The problem states $X > 3$ , so X cannot equal 3. If X were 3, one side would become 0 cm, making it impossible to form a rectangle!

Does the difference of 9 cm² stay the same for any value of X?

Yes! No matter what value X has (as long as X > 3), the square will always have exactly 9 cm² more area than the rectangle.

How do I remember the difference of squares formula?

Think of it as (First + Last)(First - Last) = First² - Last². The middle terms cancel out when you expand: $(X+3)(X-3) = X^2 - 3X + 3X - 9 = X^2 - 9$

Can this method work for other shape modifications?

Absolutely! This algebraic approach works whenever you're comparing areas with linear modifications to dimensions. Just set up the area formulas and subtract to find the difference.

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Area Comparison: Square vs. Modified Rectangle with Side X cm

Area Comparison with Algebraic Expressions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just compare X+3 and X-3 to determine which area is larger?

What if X = 3? Would the rectangle still be smaller?

Does the difference of 9 cm² stay the same for any value of X?

How do I remember the difference of squares formula?

Can this method work for other shape modifications?

🌟 Unlock Your Math Potential

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