Calculate Rectangle Area: Square with Side x Transformed by ±3cm

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

Think of it as "outer squared minus inner squared": $ (a+b)(a-b) = a^2 - b^2 $. The middle terms $ +ab $ and $ -ab $ always cancel out when you have this special pattern!

Q: What if x was less than 3?

That's why the problem states $ x > 3 $! If $ x ≤ 3 $, then $ x - 3 ≤ 0 $, which would give us a negative or zero side length - impossible for a real rectangle!

Q: Should I expand (x+3)(x-3) step by step or use the formula?

Both methods work, but using the difference of squares formula is much faster! Once you recognize the pattern $ (a+b)(a-b) $, you can immediately write $ x^2 - 9 $ without expanding.

Q: Why is the answer x² - 9 instead of x² + 9?

Because we're using multiplication, not addition! When you multiply $ (x+3) \times (x-3) $, you get $ x^2 - 3^2 = x^2 - 9 $. The rectangle area is actually smaller than the original square's area.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the rectangle using X

00:03 Draw the new rectangle according to the given data

00:12 Use the formula for calculating rectangle area (side times side)

00:22 Open parentheses correctly

00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

If the length of the side of a square is X cm

$(x>3)$

Extend one side by 3 cm and shorten an adjacent side by 3 cm in order to obtain a rectangle.

Express the area of the rectangle using x.

2

Step-by-step solution

First, let's recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$ ) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$ (square units)

With this is mind, let's proceed to solve the problem:

Calculate the area of the rectangle whose vertices we'll mark with letters $EFGH$

We are told that one side of the rectangle is obtained by extending one side of the square with side length $x$ (cm) by 3 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 3 cm:

Therefore, the lengths of the rectangle's sides are:

$EF=HG=x+3\\ EH=FG=x-3$ cm,

Apply the above formula in order to calculate the area of the rectangle that was formed from the square as described in the question:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3)$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the difference of squares formula:

$(c+d)(c-d)=c^2-d^2$ The area of the rectangle using the above formula is as follows:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+3)(x-3) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3^2\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-9}$ (sq cm)

Therefore, the correct answer is answer C.

3

Final Answer

$x^2-9\left(\operatorname{cm}²\right)$

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width (A = l × w)
Technique: Apply difference of squares: $(x+3)(x-3) = x^2 - 9$
Check: Verify dimensions make sense: both $x+3$ and $x-3$ are positive when x > 3 ✓

Common Mistakes

Avoid these frequent errors

Expanding incorrectly or forgetting the difference of squares formula
Don't expand $(x+3)(x-3)$ by distributing each term = $x^2 + 6x - 9$ which is wrong! This misses the special pattern where middle terms cancel out. Always recognize difference of squares: $(a+b)(a-b) = a^2 - b^2$ .

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

Why does extending one side and shortening another change the area?

+

Even though you're adding and subtracting the same amount (3 cm), the multiplication effect is different! The area becomes $(x+3)(x-3) = x^2 - 9$ , which is actually smaller than the original square's area of $x^2$ .

How do I remember the difference of squares formula?

+

Think of it as "outer squared minus inner squared": $(a+b)(a-b) = a^2 - b^2$ . The middle terms $+ab$ and $-ab$ always cancel out when you have this special pattern!

What if x was less than 3?

+

That's why the problem states $x > 3$ ! If $x ≤ 3$ , then $x - 3 ≤ 0$ , which would give us a negative or zero side length - impossible for a real rectangle!

Should I expand (x+3)(x-3) step by step or use the formula?

+

Both methods work, but using the difference of squares formula is much faster! Once you recognize the pattern $(a+b)(a-b)$ , you can immediately write $x^2 - 9$ without expanding.

Why is the answer x² - 9 instead of x² + 9?

+

Because we're using multiplication, not addition! When you multiply $(x+3) \times (x-3)$ , you get $x^2 - 3^2 = x^2 - 9$ . The rectangle area is actually smaller than the original square's area.

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Calculate Rectangle Area: Square with Side x Transformed by ±3cm

Rectangle Area with Side Modifications

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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 does extending one side and shortening another change the area?

How do I remember the difference of squares formula?

What if x was less than 3?

Should I expand (x+3)(x-3) step by step or use the formula?

Why is the answer x² - 9 instead of x² + 9?

🌟 Unlock Your Math Potential

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