Rewriting (x^2+4)(y^2+4) As A Sum Of Two Squares

Hey guys! Let's dive into a fun mathematical problem: rewriting the expression $(x^2+4)(y^2+4)$ using the sum of two squares. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. We'll explore the options given and figure out which one correctly represents the original expression in the form of $A^2 + B^2$. So, let's get started and unravel this mathematical puzzle together!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We have the expression $(x^2 + 4)(y^2 + 4)$, and our goal is to rewrite it in the form of a sum of two squares, which looks like $A^2 + B^2$. This means we need to find two expressions, let's call them A and B, such that when we square them and add them together, we get the same result as $(x^2 + 4)(y^2 + 4)$.

Why is this useful? Well, rewriting expressions in different forms can help us simplify them, solve equations, or even understand the underlying mathematical relationships better. Plus, it's a great exercise for our algebraic muscles! So, let's get those muscles flexing.

Key Concepts to Remember

• Sum of Two Squares: This refers to an expression in the form $A^2 + B^2$, where A and B are algebraic terms.
• Expansion: We'll need to expand the given expression $(x^2 + 4)(y^2 + 4)$ to see what it looks like in expanded form. This involves multiplying each term in the first set of parentheses by each term in the second set.
• Algebraic Manipulation: We'll be using algebraic techniques to rearrange terms and try to fit the expanded expression into the sum of two squares format.

Initial Expansion

First, let’s expand the given expression $(x^2 + 4)(y^2 + 4)$:

$(x^2 + 4)(y^2 + 4) = x^2y^2 + 4x^2 + 4y^2 + 16$

Now, we need to figure out how to rewrite this expanded form as a sum of two squares. This is where the options provided come into play. We'll examine each option to see if it matches our expanded expression.

Analyzing the Options

Okay, now let's take a look at the options provided and see which one fits the bill. We'll expand each option and compare it to the expanded form of our original expression, which we found to be $x^2y^2 + 4x^2 + 4y^2 + 16$.

Option A: $(x y-16)^2-(4 x+4 y)^2$

Let's expand this option:

$(x y - 16)^2 - (4x + 4y)^2 = (x^2y^2 - 32xy + 256) - (16x^2 + 32xy + 16y^2)$ $= x^2y^2 - 32xy + 256 - 16x^2 - 32xy - 16y^2$ $= x^2y^2 - 16x^2 - 16y^2 - 64xy + 256$

This doesn't match our expanded original expression, so Option A is not the correct answer.

Option B: $(x y-2)^2-(2 x+2 y)^2$

Expanding this option:

$(x y - 2)^2 - (2x + 2y)^2 = (x^2y^2 - 4xy + 4) - (4x^2 + 8xy + 4y^2)$ $= x^2y^2 - 4xy + 4 - 4x^2 - 8xy - 4y^2$ $= x^2y^2 - 4x^2 - 4y^2 - 12xy + 4$

Again, this doesn't match our expanded original expression, so Option B is also incorrect.

Option C: $(2 x y+4)^2+(2 x+2 y)^2$

Expanding this option:

Oops! It seems there's a small mistake in the original options provided. Option C should likely be $(xy - 4)^2 + (2x + 2y)^2$ to align with the sum of two squares format and the given expression. However, let’s continue expanding Option C as it's written and then correct it.

$(2xy + 4)^2 + (2x + 2y)^2 = (4x^2y^2 + 16xy + 16) + (4x^2 + 8xy + 4y^2)$ $= 4x^2y^2 + 4x^2 + 4y^2 + 24xy + 16$

This also doesn’t match our original expanded expression.

Option D: $(x y-4)^2+(2 x+2 y)^2$

Expanding this option:

$(x y - 4)^2 + (2x + 2y)^2 = (x^2y^2 - 8xy + 16) + (4x^2 + 8xy + 4y^2)$ $= x^2y^2 - 8xy + 16 + 4x^2 + 8xy + 4y^2$ $= x^2y^2 + 4x^2 + 4y^2 + 16$

Hey, bingo! This option perfectly matches our expanded original expression: $x^2y^2 + 4x^2 + 4y^2 + 16$. So, Option D is the correct answer!

The Correct Answer and Why

The correct answer is D. $(x y-4)^2+(2 x+2 y)^2$.

We arrived at this answer by expanding both the original expression and each of the options. By comparing the expanded forms, we found that Option D perfectly matched the expanded form of $(x^2 + 4)(y^2 + 4)$, which is $x^2y^2 + 4x^2 + 4y^2 + 16$.

Euler's Four-Square Identity

This problem is a classic example of a concept related to Euler's four-square identity, which is a more general identity that deals with the product of sums of four squares. In our case, we're dealing with a simplified version that involves sums of two squares, but the underlying principle is similar.

The identity we've effectively used here can be written as:

$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$

If we let $a = x$, $b = 2$, $c = y$, and $d = 2$, we can see how this identity applies to our problem:

$(x^2 + 2^2)(y^2 + 2^2) = (xy - 2*2)^2 + (x*2 + 2*y)^2$ $(x^2 + 4)(y^2 + 4) = (xy - 4)^2 + (2x + 2y)^2$

This identity provides a neat way to rewrite the product of sums of squares as another sum of squares. Pretty cool, right?

Alternative Approaches (Optional)

While expanding and comparing was a straightforward way to solve this problem, there are other approaches we could have taken. Here are a couple of alternative methods:

1. Working Backwards: Instead of expanding the options, we could have tried to manipulate the expanded form of the original expression ($x^2y^2 + 4x^2 + 4y^2 + 16$) directly into the sum of two squares format. This would involve some clever algebraic manipulation and pattern recognition.

2. Using Complex Numbers: This approach is a bit more advanced, but it's worth mentioning. We can rewrite $x^2 + 4$ as $x^2 + (2i)(-2i)$ and $y^2 + 4$ as $y^2 + (2i)(-2i)$, where i is the imaginary unit ($\sqrt{-1}$). Then, we can use properties of complex numbers to rewrite the product as a sum of squares.

These alternative approaches might be a bit more challenging, but they offer different perspectives on the problem and can help you develop a deeper understanding of algebraic concepts.

Tips and Tricks for Similar Problems

Okay, so we've solved this problem, but what if you encounter a similar one in the future? Here are some tips and tricks to keep in mind:

• Expand Carefully: When dealing with algebraic expressions, especially those involving squares and parentheses, it's crucial to expand carefully. Double-check your work to avoid making mistakes with signs or terms.
• Look for Patterns: Many algebraic identities and manipulations rely on recognizing patterns. Practice helps you become more adept at spotting these patterns.
• Consider Different Forms: Remember that there's often more than one way to write an algebraic expression. Being able to switch between different forms (e.g., expanded form, factored form, sum of squares) is a valuable skill.
• Don't Be Afraid to Experiment: If you're not sure how to proceed, try something! Experiment with different manipulations and see if they lead you closer to the solution. Sometimes, the best way to learn is by trying things out.
• Use Identities: Familiarize yourself with common algebraic identities like the difference of squares, perfect square trinomials, and, as we saw in this problem, Euler's four-square identity (or its simpler variants).

Conclusion

So, there you have it! We successfully rewrote the expression $(x^2 + 4)(y^2 + 4)$ as a sum of two squares, and we did it together. We broke down the problem, expanded the expressions, analyzed the options, and even explored some alternative approaches and helpful tips. Give yourselves a pat on the back – you've earned it!

Remember, math isn't about memorizing formulas; it's about understanding concepts and developing problem-solving skills. By working through problems like this one, you're not just learning math; you're learning how to think critically and creatively. Keep practicing, keep exploring, and keep having fun with math!

If you have any questions or want to explore more problems like this, feel free to ask. Until next time, keep those algebraic muscles flexing!