Simplify: $(3√y + √x)(2√y - √x)$ Explained

Let's dive into expanding and simplifying the expression $(3 \sqrt{y} + \sqrt{x})(2 \sqrt{y} - \sqrt{x})$. This type of problem often appears in algebra and calculus, so mastering it is super useful. Guys, get ready to break it down step by step!

Step-by-Step Expansion

To expand this expression, we'll use the FOIL method (First, Outer, Inner, Last). This helps ensure we multiply each term in the first parenthesis by each term in the second parenthesis systematically.

1. First: Multiply the first terms in each parenthesis:

   $(3 \sqrt{y}) * (2 \sqrt{y}) = 6y$

   Explanation: When you multiply $3 \sqrt{y}$ by $2 \sqrt{y}$, you multiply the coefficients (3 and 2) to get 6, and you multiply the square roots ($\sqrt{y} * \sqrt{y}$) to get $y$. So, $3 * 2 * \sqrt{y} * \sqrt{y} = 6y$.

2. Outer: Multiply the outer terms:

   $(3 \sqrt{y}) * (- \sqrt{x}) = -3 \sqrt{xy}$

   Explanation: Here, you multiply $3 \sqrt{y}$ by $- \sqrt{x}$. The result is $-3$ times the square root of $x$ times $y$, which is written as $-3 \sqrt{xy}$.

3. Inner: Multiply the inner terms:

   $(\sqrt{x}) * (2 \sqrt{y}) = 2 \sqrt{xy}$

   Explanation: You multiply $\sqrt{x}$ by $2 \sqrt{y}$. This gives you $2$ times the square root of $x$ times $y$, written as $2 \sqrt{xy}$.

4. Last: Multiply the last terms in each parenthesis:

   $(\sqrt{x}) * (- \sqrt{x}) = -x$

   Explanation: When you multiply $\sqrt{x}$ by $- \sqrt{x}$, you get $-x$ because $\sqrt{x} * \sqrt{x} = x$, and then you apply the negative sign.

Now, let's put it all together:

$6y - 3 \sqrt{xy} + 2 \sqrt{xy} - x$

Simplifying the Expression

After expanding, we look for like terms to simplify the expression. In this case, we have two terms with $\sqrt{xy}$:

$-3 \sqrt{xy}$ and $2 \sqrt{xy}$

Combine these like terms:

$-3 \sqrt{xy} + 2 \sqrt{xy} = - \sqrt{xy}$

So, the simplified expression is:

$6y - \sqrt{xy} - x$

Final Result

Therefore, $(3 \sqrt{y} + \sqrt{x})(2 \sqrt{y} - \sqrt{x})$ simplifies to $6y - \sqrt{xy} - x$. This is your final simplified expression. Remember, guys, always double-check your work to ensure accuracy!

Common Mistakes to Avoid

When working with expressions like these, it's easy to make a few common mistakes. Here are some things to watch out for:

• Incorrectly Applying the Distributive Property: Make sure every term in the first parenthesis is multiplied by every term in the second parenthesis.
• Mistakes with Square Roots: Remember that $\sqrt{a} * \sqrt{b} = \sqrt{ab}$, but you can only combine terms under the square root if they are being multiplied or divided, not added or subtracted.
• Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, $3 \sqrt{xy}$ and $2 \sqrt{xy}$ can be combined because they both contain $\sqrt{xy}$, but $6y$ and $-x$ cannot be combined because they are different variables.
• Sign Errors: Pay close attention to negative signs, especially when distributing them. A simple sign error can throw off the entire problem.

Practice Problems

To solidify your understanding, try these practice problems:

1. $(2 \sqrt{a} + \sqrt{b})(3 \sqrt{a} - \sqrt{b})$
2. $(4 \sqrt{x} - \sqrt{y})( \sqrt{x} + 2 \sqrt{y})$
3. $(\sqrt{p} + 3 \sqrt{q})( \sqrt{p} - 3 \sqrt{q})$

Work through these problems, and then check your answers with a friend or teacher. Practice makes perfect, guys!

Real-World Applications

While simplifying expressions might seem like an abstract math exercise, it has real-world applications in various fields, such as:

• Physics: Simplifying equations in mechanics and electromagnetism.
• Engineering: Calculating stress and strain in structural analysis.
• Computer Graphics: Manipulating equations for transformations and rendering.

Understanding how to manipulate and simplify expressions is a fundamental skill that can help you in many areas of study and work. It's one of those skills that, once mastered, opens doors to more complex and interesting problems.

Conclusion

Alright, guys, we've walked through how to expand and simplify the expression $(3 \sqrt{y} + \sqrt{x})(2 \sqrt{y} - \sqrt{x})$. Remember to use the FOIL method, combine like terms carefully, and watch out for those common mistakes. Keep practicing, and you'll become a pro at simplifying expressions in no time! This skill will not only help you in math class but also in various real-world applications. So keep up the great work and happy simplifying!

Mastering these algebraic manipulations is essential for anyone looking to excel in math and science. Whether you're a student preparing for an exam or a professional tackling complex calculations, the ability to quickly and accurately simplify expressions is invaluable. Don't underestimate the power of practice, and always strive to understand the underlying principles. Remember, a solid foundation in algebra can take you far in your academic and professional pursuits.

So, next time you encounter an expression like $(3 \sqrt{y} + \sqrt{x})(2 \sqrt{y} - \sqrt{x})$, you'll be well-equipped to tackle it with confidence. Keep honing your skills, and you'll find that even the most daunting mathematical challenges become manageable. And remember, if you ever get stuck, don't hesitate to seek help from teachers, classmates, or online resources. The journey of learning math is a collaborative one, and there's always someone willing to lend a hand.

Keep exploring, keep questioning, and keep pushing your boundaries. The world of mathematics is vast and fascinating, and there's always something new to discover. So embrace the challenge, and enjoy the process of learning and growing. With dedication and perseverance, you can achieve anything you set your mind to. Now, go forth and conquer those algebraic expressions!