Find the Area Expression: Square with Side Length (x+1)

Q: Why can't the area just be x(x+1)?

That would be the area of a rectangle with sides x and (x+1). But this is a square, so both sides are the same length: (x+1). The area must be $ (x+1)^2 $.

Q: How do I remember the perfect square formula?

Think FOIL: First + Outer + Inner + Last. For $ (x+1)^2 $, you get $ x^2 + x + x + 1 = x^2 + 2x + 1 $. The middle term is always twice the product of the two terms.

Q: What if I chose (x+1)(x-1) as my answer?

That's a common mix-up! $ (x+1)(x-1) $ is the difference of squares pattern, which gives $ x^2 - 1 $. But our square has both sides equal to (x+1).

Q: How can I check if my expanded form is correct?

Pick a simple value like x = 3. Then $ (3+1)^2 = 16 $ and $ 3^2 + 2(3) + 1 = 9 + 6 + 1 = 16 $. If they match, you're right!

Area Formulas with Binomial Expansion

Choose the expression that represents the area of the square below.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the square

00:03 We'll use the formula for calculating the area of a square (side squared)

00:10 We'll use the shortened multiplication formulas to expand the brackets

00:15 We'll calculate the multiplication and the square

00:24 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that represents the area of the square below.

Step-by-step solution

First, let's recall the formula for calculating the area of a square with side length y (length units):

$S_{\boxed{}}=y^2$ Therefore, for a square with side length:

$x+1$ (length units), the expression for the area is:

$S_{\boxed{}}=(x+1)^2$ Now, in order to simplify the expression, let's recall the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$ Let's continue and apply this formula to the area expression we got:

$S_{\boxed{}}=(x+1)^2 \\ \downarrow\\ S_{\boxed{}}=x^2+2\cdot x\cdot 1+1\\ \boxed{ S_{\boxed{}}=x^2+2x+1}\\$ This is the most simplified expression for the given square's area,

therefore the correct answer is answer D.

Final Answer

$x^2+2x+1$

Key Points to Remember

Essential concepts to master this topic

Square Area Rule: Area equals side length squared: $A = s^2$
Perfect Square Formula: $(x+1)^2 = x^2 + 2x + 1$ using $(a+b)^2 = a^2 + 2ab + b^2$
Verification Check: Substitute a value like x=2: $(2+1)^2 = 9$ and $2^2 + 2(2) + 1 = 9$ ✓

Common Mistakes

Avoid these frequent errors

Calculating area as side times side without expanding
Don't write (x+1)(x+1) and stop there = incomplete answer! The question asks for the simplified expression. Always expand the binomial using the perfect square formula to get $x^2 + 2x + 1$ .

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

Why can't the area just be x(x+1)?

That would be the area of a rectangle with sides x and (x+1). But this is a square, so both sides are the same length: (x+1). The area must be $(x+1)^2$ .

How do I remember the perfect square formula?

Think FOIL: First + Outer + Inner + Last. For $(x+1)^2$ , you get $x^2 + x + x + 1 = x^2 + 2x + 1$ . The middle term is always twice the product of the two terms.

What if I chose (x+1)(x-1) as my answer?

That's a common mix-up! $(x+1)(x-1)$ is the difference of squares pattern, which gives $x^2 - 1$ . But our square has both sides equal to (x+1).

How can I check if my expanded form is correct?

Pick a simple value like x = 3. Then $(3+1)^2 = 16$ and $3^2 + 2(3) + 1 = 9 + 6 + 1 = 16$ . If they match, you're right!

Do I always need to expand the area expression?

It depends on what the question asks for! If it asks for the simplified or expanded form, then yes. Always read the question carefully to see what form of answer is wanted.

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