Find Equivalent Expressions for (x+y)²: Perfect Square Expansion

Q: Why isn't $ (x+y)^2 $ just $ x^2 + y^2 $ ?

Because squaring a binomial means multiplying it by itself: $ (x+y)(x+y) $. When you FOIL this, you get four terms: $ x^2 + xy + yx + y^2 $, which combines to $ x^2 + 2xy + y^2 $.

Q: How do I remember the perfect square formula?

Think "First, Twice, Last": First term squared + Twice the product of both terms + Last term squared. For $ (x+y)^2 $: $ x^2 $ + $ 2xy $ + $ y^2 $!

Q: What's the difference between $ (x+y)^2 $ and $ (x-y)^2 $ ?

The middle term sign changes! $ (x+y)^2 = x^2 + 2xy + y^2 $ has plus $ 2xy $, while $ (x-y)^2 = x^2 - 2xy + y^2 $ has minus $ 2xy $.

Q: Can I rearrange the terms in the answer?

Yes! Addition is commutative, so $ x^2 + 2xy + y^2 $, $ y^2 + x^2 + 2xy $, and $ 2xy + x^2 + y^2 $ are all equivalent expressions.

Q: How can I check my expansion is correct?

Try substituting simple numbers! If $ x = 1 $ and $ y = 2 $, then $ (1+2)^2 = 9 $. Your expanded form should also equal 9: $ 1^2 + 2(1)(2) + 2^2 = 1 + 4 + 4 = 9 $ ✓

Perfect Square Binomials with Middle Term Recognition

Choose the expression that has the same value as the following:

$(x+y)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the expression

00:03 Every factor squared is actually its multiplication by itself

00:10 Let's properly open the parentheses, multiply each factor by each factor

00:33 Collect like terms

00:45 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 has the same value as the following:

$(x+y)^2$

Step-by-step solution

To solve this problem, we will use the formula for the square of a sum, which is $(a+b)^2 = a^2 + 2ab + b^2$ .

Here, we identify $a = x$ and $b = y$ . Thus, our expression $(x+y)^2$ can be expanded as follows:

$(x+y)^2 = x^2 + 2xy + y^2$

Now we will match this expanded form with the given choices.

Choice 1: $x^2 + y^2$ - This is missing the middle term $2xy$ .
Choice 2: $y^2 + x^2 + 2xy$ - This matches the expanded expression.
Choice 3: $x^2 + xy + y^2$ - The term $xy$ doesn't match the required $2xy$ .
Choice 4: $x^2 - 2xy + y^2$ - The middle term has an incorrect sign.

Therefore, the expression that is equivalent to $(x+y)^2$ is choice 2: $y^2 + x^2 + 2xy$ .

Final Answer

$y^2+x^2+2xy$

Key Points to Remember

Essential concepts to master this topic

Formula: $(a+b)^2 = a^2 + 2ab + b^2$ for any binomial square
Technique: Identify $a = x$ and $b = y$ to get $x^2 + 2xy + y^2$
Check: Count three terms with middle term coefficient being $2xy$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term when expanding perfect squares
Don't expand $(x+y)^2$ as just $x^2 + y^2$ = missing the $2xy$ term! This ignores the cross-multiplication that happens when FOILing $(x+y)(x+y)$ . Always remember the formula produces three terms: first squared + twice the product + second squared.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why isn't $(x+y)^2$ just $x^2 + y^2$ ?

Because squaring a binomial means multiplying it by itself: $(x+y)(x+y)$ . When you FOIL this, you get four terms: $x^2 + xy + yx + y^2$ , which combines to $x^2 + 2xy + y^2$ .

How do I remember the perfect square formula?

Think "First, Twice, Last": First term squared + Twice the product of both terms + Last term squared. For $(x+y)^2$ : $x^2$ + $2xy$ + $y^2$ !

What's the difference between $(x+y)^2$ and $(x-y)^2$ ?

The middle term sign changes! $(x+y)^2 = x^2 + 2xy + y^2$ has plus $2xy$ , while $(x-y)^2 = x^2 - 2xy + y^2$ has minus $2xy$ .

Can I rearrange the terms in the answer?

Yes! Addition is commutative, so $x^2 + 2xy + y^2$ , $y^2 + x^2 + 2xy$ , and $2xy + x^2 + y^2$ are all equivalent expressions.

How can I check my expansion is correct?

Try substituting simple numbers! If $x = 1$ and $y = 2$ , then $(1+2)^2 = 9$ . Your expanded form should also equal 9: $1^2 + 2(1)(2) + 2^2 = 1 + 4 + 4 = 9$ ✓

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Find Equivalent Expressions for (x+y)²: Perfect Square Expansion

Perfect Square Binomials with Middle Term Recognition

❤️ 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 isn't $(x+y)^2$ just $x^2 + y^2$ ?

How do I remember the perfect square formula?

What's the difference between $(x+y)^2$ and $(x-y)^2$ ?

Can I rearrange the terms in the answer?

How can I check my expansion is correct?

🌟 Unlock Your Math Potential

More Questions

Square of sum

Find Equivalent Expressions for (x+3)²: Perfect Square Expansion

Solve (x+1)² + (x+2)²: Sum of Squared Binomial Expressions

Solve Rational Expression: (2x+1)²/(x+2) + (x+2)²/(2x+1) = 4.5x

Solve the System: x²+4=-6y and y²+9=-4x Equations

Find Equivalent Expressions for (x+y)²: Perfect Square Expansion

Perfect Square Binomials with Middle Term Recognition

❤️ 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 isn't (x+y)2 (x+y)^2 (x+y)2 just x2+y2 x^2 + y^2 x2+y2?

How do I remember the perfect square formula?

What's the difference between (x+y)2 (x+y)^2 (x+y)2 and (x−y)2 (x-y)^2 (x−y)2?

Can I rearrange the terms in the answer?

How can I check my expansion is correct?

🌟 Unlock Your Math Potential

More Questions

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Find Equivalent Expressions for (x+3)²: Perfect Square Expansion

Solve (x+1)² + (x+2)²: Sum of Squared Binomial Expressions

Solve Rational Expression: (2x+1)²/(x+2) + (x+2)²/(2x+1) = 4.5x

Solve the System: x²+4=-6y and y²+9=-4x Equations

Why isn't $(x+y)^2$ just $x^2 + y^2$ ?

What's the difference between $(x+y)^2$ and $(x-y)^2$ ?