Justifying Step 3: (x+y)^2-(x-y)^2=4xy Proof Explained

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Justifying Step 3 in the Proof of the Identity (x+y)^2-(x-y)^2=4xy

Hey guys! Let's dive into a cool math problem today. We're going to break down a proof and figure out what makes each step tick. Specifically, we're looking at the identity (x+y)2-(x-y)2=4xy. Imagine Darlene wrote out a proof for this, and we need to understand why she did what she did, especially for Step 3. Let’s get started!

Understanding the Initial Steps

Before we zoom in on Step 3, let's quickly recap the first couple of steps. This will give us the context we need to really grasp what's going on. Darlene's proof starts like this:

Step 1: (x+y)2-(x-y)2=(x+y)(x+y)-(x-y)(x-y)

This step is pretty straightforward. All Darlene did here was rewrite the squares using their definition. Remember, squaring something just means multiplying it by itself. So, (x+y)^2 becomes (x+y)(x+y), and (x-y)^2 becomes (x-y)(x-y). This is a crucial first step because it sets us up to expand these expressions. It's like laying the foundation for the rest of the proof. Without this step, we couldn't move forward and simplify the equation. Think of it as unpacking the squares into a form we can actually work with. It's a simple move, but it's essential for what comes next. This step directly applies the definition of an exponent, which is a fundamental concept in algebra. Making sure we understand this basic principle will help us follow the rest of the proof more easily. So far, so good!

Step 2: (x+y)(x+y)-(x-y)(x-y)=(x2+xy+xy+y2)-(x2-xy-xy+y2)

Here, Darlene has expanded the products. This means she's multiplied out the (x+y)(x+y) and (x-y)(x-y) terms. Let's break this down a bit further. When you multiply (x+y) by (x+y), you're using the distributive property (or the FOIL method, if you've heard of that). This gives you:

  • x * x = x^2
  • x * y = xy
  • y * x = xy (which is the same as xy)
  • y * y = y^2

So, (x+y)(x+y) becomes x^2 + xy + xy + y^2. We do the same thing for (x-y)(x-y):

  • x * x = x^2
  • x * -y = -xy
  • -y * x = -xy
  • -y * -y = y^2

Which makes (x-y)(x-y) equal to x^2 - xy - xy + y^2. Notice the careful use of signs here – a negative times a negative gives a positive! This expansion is a key algebraic skill. It's all about carefully multiplying each term in the first set of parentheses by each term in the second set. Get comfortable with this, and you'll be able to tackle more complex algebraic problems with confidence. The distributive property is a cornerstone of algebra, so mastering it is super important.

Focusing on Step 3 and Its Justification

Now, let's get to the heart of the matter: Step 3. The actual Step 3 isn't given in the prompt, but let's infer what it likely is based on the previous steps and the identity we're trying to prove. Given Step 2, a logical Step 3 would involve simplifying the expression by combining like terms and dealing with the subtraction. So, here's what Step 3 probably looks like:

Step 3: (x2+xy+xy+y2)-(x2-xy-xy+y2) = x^2 + 2xy + y^2 - x^2 + 2xy - y^2

What Darlene likely did here is two things:

  1. Combined like terms within each set of parentheses: In the first set of parentheses, xy + xy was combined to give 2xy. Similarly, in the second set, -xy - xy was combined to give -2xy. But because of the subtraction in front of the parenthesis it became +2xy in the step 3. This is a standard simplification technique. It's all about making the expression cleaner and easier to work with.
  2. Distributed the negative sign: The minus sign in front of the second set of parentheses means we're subtracting the entire expression inside. To do this correctly, we need to distribute the negative sign to each term inside the parentheses. This means:
    • -(x^2) becomes -x^2
    • -(-2xy) becomes +2xy
    • -(+y^2) becomes -y^2

This distribution of the negative sign is absolutely crucial. It's a very common place for mistakes, so you've got to be extra careful with it. Remember, subtracting a negative is the same as adding a positive. Getting the signs right is key to getting the whole problem right. So, the main justification for Step 3 is the combination of like terms and the correct distribution of the negative sign. This step bridges the gap between the expanded form and the simplified form, bringing us closer to the final answer. It demonstrates a solid understanding of algebraic manipulation.

The Grand Finale: Steps 4 and 5 (Anticipated)

To complete the picture, let's imagine the final steps of the proof:

Step 4: x^2 + 2xy + y^2 - x^2 + 2xy - y^2 = 4xy

In this step, Darlene cancels out the terms. Notice that we have:

  • x^2 and -x^2, which cancel each other out.
  • y^2 and -y^2, which also cancel each other out.

This leaves us with 2xy + 2xy, which simplifies to 4xy. Cancelling out terms like this is a fundamental simplification technique in algebra. It's all about identifying terms that are opposites and eliminating them. This helps to distill the expression down to its simplest form. In this case, it neatly reveals the 4xy that we're trying to prove.

Step 5: 4xy = 4xy

This final step simply states the result. We've shown that by expanding and simplifying the left side of the original equation, we arrive at 4xy, which is equal to the right side. This confirms the identity. It's like the final flourish in a mathematical argument. We've taken the initial equation, manipulated it step-by-step, and arrived at a clear and undeniable conclusion. This is what a mathematical proof is all about – demonstrating the truth of a statement through logical steps.

In Conclusion

So, there you have it! Step 3 in Darlene's proof is justified by the combination of like terms and the careful distribution of the negative sign. It's a pivotal step that sets the stage for the final simplification and the triumphant conclusion of the proof. Remember, guys, math proofs are like puzzles. Each step is a piece, and understanding the justification for each step helps you see the whole picture. Keep practicing, and you'll become a math whiz in no time! High-quality content like this is all about breaking down complex concepts into manageable chunks, making learning fun and engaging. We've covered a lot today, from the initial expansion to the final cancellation of terms. But hopefully, you now have a clearer understanding of the algebraic principles at play and why each step is important. Now go forth and conquer those math problems!