Calculate Area: Square with Side (x+1) Transforming to Rectangle

Q: What does the distributive property do here?

The distributive property helps us expand $ (x+2) \times x $. Using $ (m+n)d = md + nd $, we get: $ x(x+2) = x^2 + 2x $.

Q: Why is x > 3 given in the problem?

This constraint ensures all measurements are positive. Since we subtract 1 from $ (x+1) $ to get side length $ x $, we need $ x > 0 $. The condition $ x > 3 $ gives us a reasonable size rectangle.

Q: Can I check my answer with specific numbers?

Absolutely! Try $ x = 4 $: Original square has side 5 cm. Rectangle has sides 6 cm and 4 cm. Area = $ 6 \times 4 = 24 $ cm². Using our formula: $ 4^2 + 2(4) = 16 + 8 = 24 $ cm² ✓

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

00:14 Calculate the area of the rectangle (side multiplied by side)

00:18 Substitute appropriate values and solve to find the rectangle's area

00:24 Collect terms

00:31 Pay attention to properly closing parentheses

00:35 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

The length of the side of the square is $x+1$ cm

$(x>3)$

If we extend one side by 1 cm and shorten an adjacent side by 1 cm, we obtain a rectangle

Determine the area of the rectangle?

2

Step-by-step solution

First, 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)

Proceed to solve the problem:

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

It is given in the problem that one side of the rectangle is obtained by extending one side of the square with side length $x +1$ (cm) by 1 cm, and the second side of the rectangle is obtained by shortening the adjacent side of the given square by 1 cm:

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

$EF=HG=(x+1)+1\\ \downarrow\\ \boxed{ EF=HG=x+2}\\ \hspace{2pt}\\ \\ EH=FG=(x+1)-1\\ \downarrow\\ \boxed{ EH=FG=x }$ (cm)

Apply the above formula to calculate the area of the rectangle that was formed from the square in the way described in the problem:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=EF\cdot EH\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x+2)x$ (sq cm)

Continue to simplify the expression that we obtained for the rectangle's area, using the distributive property:

$(m+n)d=md+nd$ Therefore, applying the distributive property, we obtain the following area for the rectangle:

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

The correct answer is answer B.

3

Final Answer

$x^2+2x$

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle area equals length times width ( $a \times b$ )
Technique: Transform square side $(x+1)$ to rectangle sides: $(x+2)$ and $x$
Check: Substitute $x = 4$ : $6 \times 4 = 24$ matches $4^2 + 2(4) = 24$ ✓

Common Mistakes

Avoid these frequent errors

Using the original square side length for rectangle calculation
Don't use $(x+1) \times (x+1)$ for rectangle area = $x^2 + 2x + 1$ ! This ignores the transformation described in the problem. Always apply the +1 and -1 changes to get sides $(x+2)$ and $x$ , then multiply for area $x^2 + 2x$ .

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 do we add 1 to one side and subtract 1 from the other?

+

The problem describes transforming a square into a rectangle by extending one side by 1 cm and shortening an adjacent side by 1 cm. So from the original square side $(x+1)$ , we get rectangle sides of $(x+1)+1 = x+2$ and $(x+1)-1 = x$ .

How do I know which sides are length and width?

+

It doesn't matter which you call length or width! Since rectangle area is length × width, you can multiply $(x+2) \times x$ or $x \times (x+2)$ - both give the same result.

What does the distributive property do here?

+

The distributive property helps us expand $(x+2) \times x$ . Using $(m+n)d = md + nd$ , we get: $x(x+2) = x^2 + 2x$ .

Why is x > 3 given in the problem?

+

This constraint ensures all measurements are positive. Since we subtract 1 from $(x+1)$ to get side length $x$ , we need $x > 0$ . The condition $x > 3$ gives us a reasonable size rectangle.

Can I check my answer with specific numbers?

+

Absolutely! Try $x = 4$ : Original square has side 5 cm. Rectangle has sides 6 cm and 4 cm. Area = $6 \times 4 = 24$ cm². Using our formula: $4^2 + 2(4) = 16 + 8 = 24$ cm² ✓

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Calculate Area: Square with Side (x+1) Transforming to Rectangle

Rectangle Area with Algebraic Side Transformation

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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 do we add 1 to one side and subtract 1 from the other?

How do I know which sides are length and width?

What does the distributive property do here?

Why is x > 3 given in the problem?

Can I check my answer with specific numbers?

🌟 Unlock Your Math Potential

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