Expand $(3x-4)^2$ Made Easy

Hey guys, let's dive into the awesome world of algebra and tackle expanding expressions like $(3x-4)^2$. You know, sometimes math can look a bit intimidating, but trust me, once you get the hang of it, it's super straightforward and even kind of fun! Today, we're going to break down exactly how to simplify and expand this specific expression, $(3x-4)^2$, so you can feel confident tackling similar problems. We'll go step-by-step, making sure we understand each part of the process. No more confusion, just clear explanations and easy-to-follow steps. So, grab your notebooks, maybe a snack, and let's get started on making this algebraic expression a piece of cake!

Understanding the Expression: $(3x-4)^2$

Alright, first things first, what does $(3x-4)^2$ actually mean? In algebra, when you see a term or an expression in parentheses with a little '2' floating up in the corner, it means you need to multiply that entire thing by itself. So, $(3x-4)^2$ is the same as saying $(3x-4) * (3x-4)$. It's like saying 'take this thing and double it'. Think of it like squaring a number, say $5^2$. That means $5 * 5$, which equals 25. We're doing the exact same thing here, but instead of just a number, we have a whole little algebraic expression inside the parentheses. This concept is super important because it's the foundation for how we'll expand it. So, remember, the exponent '2' means we're squaring the entire binomial $(3x-4)$. It's not just the $3x$ or just the $-4$; it's everything inside the brackets multiplied by itself.

Methods for Expansion

Now, how do we actually go about expanding $(3x-4) * (3x-4)$? There are a couple of really handy methods, and I’ll show you both so you can pick your favorite or use whichever one makes more sense to you at the time. The first method is often called the Distributive Property, or sometimes the FOIL method (which is a handy acronym, more on that in a bit!). The second method is using the Binomial Square Formula, which is basically a shortcut derived from the distributive property. Both will get you to the same correct answer, and understanding both gives you a solid grasp of algebraic manipulation.

Method 1: The Distributive Property (or FOIL)

This is a classic and super reliable way to expand any two binomials being multiplied. When we have $(3x-4) * (3x-4)$, we need to make sure that every term in the first set of parentheses gets multiplied by every term in the second set of parentheses. Let's visualize this. We have $3x$ and $-4$ in the first set, and $3x$ and $-4$ in the second.

• First: Multiply the first terms in each binomial. That's $(3x) * (3x)$. Easy enough, right? $3 * 3$ is $9$, and $x * x$ is $x^2$. So, the first term is $9x^2$.
• Outer: Multiply the outer terms. That's $(3x) * (-4)$. $3 * -4$ is $-12$, and we still have the $x$. So, this gives us $-12x$.
• Inner: Multiply the inner terms. That's $(-4) * (3x)$. Again, $-4 * 3$ is $-12$, and we have the $x$. So, this gives us another $-12x$.
• Last: Multiply the last terms in each binomial. That's $(-4) * (-4)$. Remember, a negative times a negative is a positive! So, $-4 * -4$ equals $+16$.

Now, we've done all the multiplications. We have $9x^2$, $-12x$, $-12x$, and $+16$. The final step is to combine like terms. In this case, the like terms are the ones with $x$ in them: $-12x$ and $-12x$. If we add those together, we get $-24x$. So, putting it all together, our expanded expression is $9x^2 - 24x + 16$.

See? Not too shabby! The FOIL acronym (First, Outer, Inner, Last) is just a mnemonic to help you remember to do all four multiplications. It's a fantastic tool for expanding binomials.

Method 2: The Binomial Square Formula

This method is like a super-powered shortcut. For any expression in the form $(a+b)^2$ or $(a-b)^2$, there's a specific formula you can use. For $(a-b)^2$, the formula is: $(a-b)^2 = a^2 - 2ab + b^2$. It looks a bit abstract, but let's see how it applies to our problem, $(3x-4)^2$.

In our case:

• 'a' is $3x$
• 'b' is $4$ (we use the positive value for 'b' and the formula already accounts for the minus sign)

Now, let's plug these into the formula $a^2 - 2ab + b^2$:

• $a^2$: This is $(3x)^2$. Remember, when you square a term with a coefficient and a variable, you square both! So, $3^2$ is $9$, and $x^2$ is $x^2$. This gives us $9x^2$.
• $-2ab$: This is $-2 * (3x) * (4)$. Let's multiply the numbers: $-2 * 3 * 4 = -24$. And we still have the $x$. So, this part is $-24x$.
• $+b^2$: This is $(4)^2$. And $4 * 4$ is $16$. So, this part is $+16$.

Putting it all together, we get $9x^2 - 24x + 16$. Wowza! The exact same result as using the distributive property, but maybe a bit quicker once you memorize the formula.

Why is Expanding Important?

So, why do we even bother expanding expressions like $(3x-4)^2$? Great question, guys! Expanding is a fundamental skill in algebra because it helps us simplify complex equations and inequalities, solve for unknown variables, and prepare for more advanced topics like graphing quadratic functions. When you expand an expression, you're essentially changing its form from a compact, squared binomial into a standard polynomial form (like $ax^2 + bx + c$). This standard form is incredibly useful for many reasons. For instance, when you're trying to find the roots of a quadratic equation (where it crosses the x-axis), having it in the form $ax^2 + bx + c$ makes it much easier to apply methods like factoring or the quadratic formula. Also, in calculus, understanding how to expand expressions is crucial for differentiation and integration. It's like learning to break down a complex machine into its individual parts so you can understand how it works and how to fix it or improve it. Expanding $(3x-4)^2$ takes it from a single squared unit into its constituent terms ($9x^2$, $-24x$, and $16$), revealing its underlying structure. This skill opens doors to solving a wider range of mathematical problems and understanding mathematical concepts more deeply. It's a building block for so many other things in math and science!

Practice Makes Perfect!

Just like learning to ride a bike or play a new video game, the more you practice expanding expressions, the better and faster you'll become. Try expanding other squared binomials, like $(2x+5)^2$ or $(x-7)^2$. Use both the distributive property and the binomial square formula to see which one you prefer. Don't be afraid to make mistakes – that's how we learn! Each problem you solve strengthens your algebraic muscles. Keep at it, and soon you'll be expanding expressions like a total pro. Remember, the journey of a thousand miles begins with a single step, and mastering algebraic expansion is a fantastic step forward in your math journey. You've got this!

So there you have it, guys! Expanding $(3x-4)^2$ is totally doable. We broke it down using the distributive property (FOIL) and the binomial square formula, and both gave us the same awesome result: $9x^2 - 24x + 16$. Keep practicing, and you'll be an expansion expert in no time. Happy calculating!