Solving $(2 \sqrt{5}+5 \sqrt{2})(2 \sqrt{5}-5 \sqrt{2})$ A Detailed Explanation

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Hey guys! Today, we're diving headfirst into a fascinating mathematical problem: evaluating the expression $(2 \sqrt{5}+5 \sqrt{2})(2 \sqrt{5}-5 \sqrt{2})$. At first glance, it might seem a bit intimidating with all those square roots floating around, but trust me, we'll break it down step by step and reveal the elegant solution hiding within. So, buckle up and let's embark on this mathematical adventure together!

Recognizing the Pattern: The Key to Simplification

The secret to cracking this problem lies in recognizing a familiar algebraic pattern. If we look closely, we'll notice that the expression has the form of $(a + b)(a - b)$. This is a classic pattern known as the "difference of squares," and it has a super handy shortcut:

$(a + b)(a - b) = a^2 - b^2$

This formula is our golden ticket to simplifying the expression. By identifying the 'a' and 'b' terms in our problem, we can directly apply the formula and bypass the need for tedious multiplication. In our case, we can see that:

• $a = 2 \sqrt{5}$
• $b = 5 \sqrt{2}$

So, let's plug these values into our difference of squares formula and see the magic unfold!

Applying the Difference of Squares: A Step-by-Step Solution

Now that we've identified the pattern and know our 'a' and 'b' values, we can confidently apply the difference of squares formula:

$(2 \sqrt{5}+5 \sqrt{2})(2 \sqrt{5}-5 \sqrt{2}) = (2 \sqrt{5})^2 - (5 \sqrt{2})^2$

This step dramatically simplifies the problem. We've transformed a potentially messy multiplication problem into a straightforward subtraction of squared terms. But we're not quite there yet. We need to evaluate those squares. Let's tackle them one at a time.

First, let's square the term $2 \sqrt{5}$. Remember that squaring a term means multiplying it by itself:

$(2 \sqrt{5})^2 = (2 \sqrt{5}) * (2 \sqrt{5})$

We can rearrange and group the terms like this:

$(2 \sqrt{5}) * (2 \sqrt{5}) = 2 * 2 * \sqrt{5} * \sqrt{5}$

Now, we know that $\sqrt{5} * \sqrt{5} = 5$, so we can simplify further:

$2 * 2 * \sqrt{5} * \sqrt{5} = 4 * 5 = 20$

Great! We've successfully squared the first term. Now, let's move on to the second term, $(5 \sqrt{2})^2$. We'll follow the same steps:

$(5 \sqrt{2})^2 = (5 \sqrt{2}) * (5 \sqrt{2})$

Rearranging and grouping:

$(5 \sqrt{2}) * (5 \sqrt{2}) = 5 * 5 * \sqrt{2} * \sqrt{2}$

And since $\sqrt{2} * \sqrt{2} = 2$:

$5 * 5 * \sqrt{2} * \sqrt{2} = 25 * 2 = 50$

Fantastic! We've squared both terms. Now, we can substitute these values back into our equation:

$(2 \sqrt{5})^2 - (5 \sqrt{2})^2 = 20 - 50$

The Grand Finale: Reaching the Solution

We're in the home stretch now! All that's left is to perform the subtraction:

$20 - 50 = -30$

And there we have it! The expression $(2 \sqrt{5}+5 \sqrt{2})(2 \sqrt{5}-5 \sqrt{2})$ simplifies to -30. Isn't that neat? What started as a seemingly complex expression has been tamed by the power of algebraic patterns.

Why This Matters: The Power of Patterns in Mathematics

This problem isn't just about getting the right answer; it's about appreciating the beauty and efficiency of mathematical patterns. Recognizing the difference of squares pattern allowed us to bypass a much more cumbersome calculation. This highlights a crucial aspect of mathematics: identifying and leveraging patterns to simplify problems.

Think of it like this: Imagine you had to build a house, but you didn't know about blueprints or standard measurements. You'd be stuck figuring out every single detail from scratch, making the process incredibly slow and prone to errors. But with blueprints and standard measurements, you can follow a well-established pattern, making the construction process much faster, easier, and more reliable.

Similarly, in mathematics, recognizing patterns allows us to build upon existing knowledge and solve problems more efficiently. The difference of squares is just one example of many powerful patterns that can unlock mathematical mysteries. As you delve deeper into mathematics, you'll encounter many more of these patterns, each offering a unique shortcut to understanding and solving problems.

Beyond the Problem: Exploring Further

Now that we've conquered this problem, let's take a moment to ponder some related questions. What if we had a similar expression with different numbers or square roots? Could we still apply the difference of squares pattern? The answer is a resounding yes!

The beauty of mathematical patterns is their versatility. The difference of squares formula works for any two terms, regardless of whether they involve square roots, fractions, or even variables. As long as you have the form $(a + b)(a - b)$, you can confidently apply the formula $a^2 - b^2$.

For example, consider the expression $(3 \sqrt{7} + 2 \sqrt{3})(3 \sqrt{7} - 2 \sqrt{3})$. We can immediately recognize the difference of squares pattern with:

• $a = 3 \sqrt{7}$
• $b = 2 \sqrt{3}$

Applying the formula, we get:

$(3 \sqrt{7} + 2 \sqrt{3})(3 \sqrt{7} - 2 \sqrt{3}) = (3 \sqrt{7})^2 - (2 \sqrt{3})^2$

Following the same steps as before, we can evaluate the squares:

$(3 \sqrt{7})^2 = 9 * 7 = 63$

$(2 \sqrt{3})^2 = 4 * 3 = 12$

And finally:

$63 - 12 = 51$

So, $(3 \sqrt{7} + 2 \sqrt{3})(3 \sqrt{7} - 2 \sqrt{3}) = 51$.

This example demonstrates the power and generality of the difference of squares pattern. It's a valuable tool in your mathematical arsenal, ready to be deployed whenever you encounter expressions of this form.

Conclusion: Embracing the Elegance of Mathematics

We've successfully navigated the expression $(2 \sqrt{5}+5 \sqrt{2})(2 \sqrt{5}-5 \sqrt{2})$, revealing its elegant solution of -30. Along the way, we've not only honed our algebraic skills but also deepened our appreciation for the power of mathematical patterns.

Remember, mathematics isn't just about memorizing formulas and crunching numbers; it's about understanding the underlying principles and recognizing the connections between different concepts. By embracing patterns and seeking out shortcuts, we can unlock the beauty and efficiency of mathematics, making it a more enjoyable and rewarding endeavor.

So, keep exploring, keep questioning, and keep seeking out those mathematical patterns. You never know what amazing discoveries you might make!