Real Number Operation: Calculation And Verification

Hey guys! Let's dive into a cool math problem involving an operation defined on real numbers. We're going to explore its properties and do some calculations. Buckle up, it's gonna be a fun ride!

Understanding the Operation

First, let's define the operation. We're given that for any real numbers x and y, the operation “o” is defined as:

$x ext{ o } y = xy + 4x + 4y + 12$

This means that when we apply this operation, we multiply x and y, add 4 times x, add 4 times y, and then add 12. It looks a bit complex, but we'll break it down step by step. This operation is crucial, and understanding it thoroughly is the key to solving the problem.

Calculating $1 ext{ o } (-4) ext{ o } 3$

Step 1: Evaluate $1 ext{ o } (-4)$

To calculate $1 ext{ o } (-4)$, we substitute x = 1 and y = -4 into our operation definition:

$1 ext{ o } (-4) = (1)(-4) + 4(1) + 4(-4) + 12$

Now, let's simplify this:

$1 ext{ o } (-4) = -4 + 4 - 16 + 12$

$1 ext{ o } (-4) = -4$

So, the result of $1 ext{ o } (-4)$ is -4. It's important to be meticulous with these calculations to avoid errors.

Step 2: Evaluate $(-4) ext{ o } 3$

Now that we know $1 ext{ o } (-4) = -4$, we need to calculate $(-4) ext{ o } 3$. Again, we use the operation definition, this time with x = -4 and y = 3:

$(-4) ext{ o } 3 = (-4)(3) + 4(-4) + 4(3) + 12$

Let's simplify this expression:

$(-4) ext{ o } 3 = -12 - 16 + 12 + 12$

$(-4) ext{ o } 3 = -4$

Therefore, $1 ext{ o } (-4) ext{ o } 3 = -4$. Remember, the associativity of the operation is crucial here; it allows us to perform the operations in this order. Associativity simplifies complex calculations.

Verifying $x ext{ o } y = (x + 4)(y + 4) - 4$

Now, let's verify that $x ext{ o } y = (x + 4)(y + 4) - 4$ for any real numbers x and y. This is a key step in understanding the structure of this operation.

Step 1: Expand the right-hand side

We start by expanding the right-hand side of the equation:

$(x + 4)(y + 4) - 4 = xy + 4x + 4y + 16 - 4$

Step 2: Simplify the expanded expression

Now, let's simplify the expanded expression:

$xy + 4x + 4y + 16 - 4 = xy + 4x + 4y + 12$

Step 3: Compare with the definition of $x ext{ o } y$

Notice that the simplified expression is exactly the definition of $x ext{ o } y$:

$xy + 4x + 4y + 12 = x ext{ o } y$

Therefore, we have verified that $x ext{ o } y = (x + 4)(y + 4) - 4$ for any real numbers x and y. This alternative form of the operation is quite insightful and can be very useful in further calculations or proofs. Understanding different representations is essential in mathematics.

This form also tells us something important about the operation. It's essentially a shifted and scaled multiplication. We add 4 to both x and y, multiply them, and then subtract 4. This kind of transformation is common in various mathematical contexts and can help simplify complex problems.

The Significance of the Result

The fact that we can rewrite $x ext{ o } y$ as $(x + 4)(y + 4) - 4$ is a game-changer. It reveals the underlying structure of the operation. This form makes it easier to see how the operation behaves and can simplify further calculations. For example, if we want to solve equations involving this operation, this form can be much more manageable. Recognizing patterns and simplifying expressions are core skills in mathematics.

Let's Think Further

Now that we have a handle on this operation, let's think about some related concepts:

1. Identity Element: Is there a number e such that $x ext{ o } e = x$ for all real numbers x? Finding an identity element can simplify calculations and provide insights into the operation's behavior.
2. Inverse Element: For a given x, is there a y such that $x ext{ o } y = e$, where e is the identity element? Understanding inverses is crucial in group theory and other algebraic structures.
3. Generalizations: Can we generalize this operation to other sets or structures? What happens if we change the constants in the definition? These kinds of questions can lead to new and interesting mathematical explorations. Mathematical thinking is about extending ideas and looking for connections.

Conclusion

So, guys, we've successfully calculated $1 ext{ o } (-4) ext{ o } 3$ and verified the alternative form of the operation $x ext{ o } y$. We also touched on the importance of understanding the structure of operations and how it can simplify calculations. This journey highlights the beauty and power of mathematical thinking. Keep exploring and stay curious! This problem isn't just about numbers; it's about understanding mathematical structures and how they work.

Remember, math isn't just about formulas and calculations. It's about problem-solving, logical thinking, and discovering new patterns. Keep challenging yourself, and you'll be amazed at what you can achieve. And that's a wrap, folks! Hope you found this helpful and engaging. Until next time, keep those mathematical gears turning!