Solving Algebraic Expressions Step By Step Solutions And Explanations

Hey guys! Welcome to an exciting journey into the world of algebra, where we'll be breaking down and solving some cool expressions. Whether you're a student tackling homework, a math enthusiast looking to sharpen your skills, or just curious about the magic of algebra, this guide is for you. We'll take a step-by-step approach to unraveling these expressions, making sure you understand the "why" behind each solution. So, buckle up, grab your favorite beverage, and let's dive in!

1. X(x-7) Decoding the Distributive Property

In this section, we'll tackle the expression X(x-7). The key to solving this lies in understanding the distributive property. Think of it as sharing – the X outside the parenthesis needs to be multiplied with each term inside. This is a fundamental concept in algebra, and mastering it will make your life so much easier when dealing with more complex equations. So, how do we apply the distributive property here? Well, we take the X and multiply it by both x and -7. Let's break it down:

First, we multiply X by x, which gives us x². Remember, when you multiply a variable by itself, you're essentially squaring it. This is a basic rule of exponents that's crucial to remember. Next, we multiply X by -7, which results in -7x. It's important to pay attention to the signs here – a negative times a positive is a negative. Now, we simply combine these two results. We have x² and -7x. Since these are not like terms (one is a squared term, and the other is a linear term), we can't combine them further. They're like apples and oranges – you can't add them together directly. Therefore, our final answer is x² - 7x. See? Not too scary, right? The distributive property is like a secret weapon in algebra, and once you get the hang of it, you'll be solving expressions like a pro. Remember to always double-check your signs and make sure you've multiplied the term outside the parenthesis by every term inside. This will save you from making common mistakes and ensure you get the correct answer every time. We've successfully expanded and simplified our first expression, and that's something to be proud of. Let's keep the momentum going and move on to the next one!

2. 3x(x+6) Mastering Multiplication with Coefficients

Alright, let's jump into our next expression: 3x(x+6). This one builds upon what we learned in the previous section but adds a little twist – a coefficient! A coefficient is just a number that's multiplied by a variable, in this case, the 3 in 3x. Don't let it intimidate you, though; we'll tackle it together. Just like before, we're going to use the distributive property, but this time, we need to make sure we multiply the 3x by both terms inside the parenthesis: x and +6.

So, let's get started. First up, we multiply 3x by x. Remember, when you multiply variables with coefficients, you multiply the coefficients together and the variables together. So, 3 times an implied 1 (in front of the x) is 3, and x times x is x². This gives us 3x². Make sure you're comfortable with these basic exponent rules – they're super important in algebra. Next, we multiply 3x by +6. Again, we multiply the coefficients: 3 times 6 is 18. And we simply keep the x along for the ride. This gives us +18x. Now, just like before, we combine our results. We have 3x² and +18x. These are different terms, just like before, so we can't simplify them any further. They're not "like terms," so they can't be added together directly. Our final answer is 3x² + 18x. Fantastic! You've successfully handled an expression with a coefficient. The key takeaway here is to remember to multiply both the coefficients and the variables when using the distributive property. And always, always double-check your work, especially the signs. With a bit of practice, you'll be multiplying these expressions in your sleep. Now, let's keep the ball rolling and move on to our next challenge!

3. (x+4)(x-7) Expanding Binomials The FOIL Method

Now we're stepping up our game! Let's dive into the expression (x+4)(x-7). This is where we encounter the multiplication of two binomials (expressions with two terms). To tackle this, we'll use a handy technique called the FOIL method. FOIL stands for First, Outer, Inner, Last. It's a mnemonic device to help us remember which terms to multiply together to ensure we don't miss anything. Basically, it's a systematic way to apply the distributive property twice.

So, let's break down the FOIL method step-by-step. First, we multiply the first terms in each binomial: x from the first parenthesis and x from the second parenthesis. x times x is x². Got it? Good! Outer, we multiply the outer terms: x from the first parenthesis and -7 from the second. x times -7 is -7x. Make sure you keep that negative sign in mind. Inner, we multiply the inner terms: 4 from the first parenthesis and x from the second. 4 times x is 4x. And finally, Last, we multiply the last terms: 4 from the first parenthesis and -7 from the second. 4 times -7 is -28. Remember to pay attention to those signs! Now, we have four terms: x², -7x, 4x, and -28. The next step is to combine any like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have -7x and 4x. Combining these, we get -3x (-7 + 4 = -3). So, our final answer is x² - 3x - 28. Awesome! You've just successfully expanded two binomials using the FOIL method. Remember, FOIL is a powerful tool, but it's really just a way to systematically apply the distributive property. Keep practicing, and you'll become a binomial-multiplying master in no time. Now, let's move on to the next expression and keep challenging ourselves!

4. (x-6)(x+6) Spotting the Difference of Squares Pattern

Alright, let's tackle (x-6)(x+6). At first glance, this might seem like just another binomial multiplication problem, but there's something special about it. Notice how the two binomials are almost identical, except for the sign in the middle? This is a classic example of what we call the difference of squares pattern. Recognizing this pattern can save you a lot of time and effort because there's a shortcut! But first, let's go through the FOIL method, just to show you how it works, and then we'll reveal the shortcut.

Using the FOIL method, we start with First: x times x is x². Then, Outer: x times 6 is 6x. Next, Inner: -6 times x is -6x. And finally, Last: -6 times 6 is -36. So, we have x² + 6x - 6x - 36. Now, let's combine like terms. Notice anything interesting? We have +6x and -6x. These cancel each other out! This is the key to the difference of squares pattern. So, we're left with just x² - 36. Ta-da! Now, let's talk about the shortcut. The difference of squares pattern states that (a - b)(a + b) = a² - b². In our case, a is x and b is 6. So, we can directly apply the formula: x² - 6² = x² - 36. See how much faster that was? Recognizing patterns like this is a superpower in algebra. It allows you to bypass lengthy calculations and arrive at the answer quickly and efficiently. The difference of squares pattern is a valuable tool in your algebraic arsenal. Remember to look for it whenever you're multiplying two binomials that have the same terms but opposite signs. Now that you know this cool trick, let's move on to our final expression!

5. (x-9)(x-9) Perfect Square Trinomials Understanding the Pattern

Last but not least, we have (x-9)(x-9). This expression is another special case: the square of a binomial. You might also see it written as (x-9)². Just like the difference of squares, recognizing this pattern can save you some time and effort. This pattern results in what's called a perfect square trinomial. Let's first expand it using the FOIL method, and then we'll discuss the pattern and its shortcut.

Applying FOIL, we start with First: x times x is x². Then, Outer: x times -9 is -9x. Next, Inner: -9 times x is -9x. And finally, Last: -9 times -9 is 81. Remember, a negative times a negative is a positive! So, we have x² - 9x - 9x + 81. Now, let's combine like terms. We have -9x and -9x, which combine to -18x. So, our expanded expression is x² - 18x + 81. This is a perfect square trinomial. Now, let's talk about the pattern. The general form for squaring a binomial like this is (a - b)² = a² - 2ab + b². In our case, a is x and b is 9. Let's see how this applies. a² is x². -2ab is -2 * x * 9 = -18x. And b² is 9² = 81. See? It matches our expanded form perfectly. Recognizing the perfect square trinomial pattern allows you to skip the FOIL method and go straight to the answer. It's a valuable shortcut to have in your algebraic toolkit. The key takeaway here is to remember the formula and how it applies to the binomial you're squaring. Once you master this pattern, you'll be able to expand these expressions in a snap. And with that, we've conquered our final expression! You've done a fantastic job tackling these algebraic challenges.

Wow, guys! We've covered so much ground in this deep dive into algebraic expressions. From the distributive property to FOIL and special patterns like the difference of squares and perfect square trinomials, you've gained some serious algebraic superpowers. Remember, the key to mastering math is practice. Keep working at it, and don't be afraid to make mistakes – that's how we learn! Whether you're tackling homework, studying for a test, or just exploring the fascinating world of math, these skills will serve you well. Keep challenging yourself, and never stop learning. You've got this!