Solving $(7a + 3)(4a - 12) = 0$: A Step-by-Step Guide

Hey everyone! Let's dive into solving the equation $(7a + 3)(4a - 12) = 0$. This type of problem often appears in algebra, and understanding how to tackle it is super useful. We'll break it down step by step, so you can follow along easily. So, let’s get started and make math a little less intimidating, shall we?

Understanding the Zero Product Property

When we're faced with an equation like $(7a + 3)(4a - 12) = 0$, the first thing we need to remember is the Zero Product Property. This property is the key to unlocking the solution. Basically, it states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if we have two expressions, let’s call them A and B, and A multiplied by B equals zero (A × B = 0), then either A = 0, or B = 0, or both A and B equal zero. This might sound a bit technical, but it’s actually quite straightforward in practice.

Think of it this way: if you're multiplying two numbers and the result is zero, one of those numbers has to be zero. There’s no other way to get zero as the product! This simple concept is incredibly powerful for solving many algebraic equations, including the one we're tackling today. The Zero Product Property allows us to take a complex-looking equation and break it down into simpler, more manageable parts. By setting each factor in the equation to zero, we can create individual equations that are much easier to solve. This is the fundamental principle we'll use to find the values of a that satisfy our original equation. So, with this property in mind, let's move on to applying it to our specific problem and see how it helps us find the solutions. Remember, understanding the 'why' behind the method makes the 'how' much easier to grasp!

Applying the Zero Product Property to $(7a + 3)(4a - 12) = 0$

Alright, let's get practical and apply the Zero Product Property to our equation: $(7a + 3)(4a - 12) = 0$. Remember, the Zero Product Property tells us that if the product of two factors is zero, then at least one of the factors must be zero. In our case, the factors are $(7a + 3)$ and $(4a - 12)$. So, to solve this equation, we need to consider two possibilities:

1. $(7a + 3) = 0$
2. $(4a - 12) = 0$

By setting each factor equal to zero, we've transformed our single, more complex equation into two simpler equations. This is a crucial step because it allows us to isolate the variable a and find its possible values. Each of these smaller equations is now much easier to handle, and we can solve them independently. Think of it like breaking down a big problem into smaller, bite-sized pieces – it makes the whole task less daunting! Now, let's take a closer look at each of these equations and solve them one by one. This is where the basic algebraic techniques come into play, and we'll see how to manipulate the equations to get a by itself on one side. So, stick with me as we move on to the next step, where we'll solve these equations and uncover the values of a that make the original equation true. We're on our way to cracking this problem!

Solving the First Equation: $7a + 3 = 0$

Okay, let's tackle the first equation: $7a + 3 = 0$. Our goal here is to isolate a on one side of the equation. To do this, we'll use some basic algebraic manipulations. The first step is to get rid of the constant term, which is +3 in this case. We can do this by subtracting 3 from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to maintain the balance. So, subtracting 3 from both sides gives us:

$7a + 3 - 3 = 0 - 3$

This simplifies to:

$7a = -3$

Now, we have $7a$ on the left side, but we want just a. To get a by itself, we need to get rid of the 7 that's multiplying it. We can do this by dividing both sides of the equation by 7. Again, it's crucial to perform the same operation on both sides to keep the equation balanced:

rac{7a}{7} = rac{-3}{7}

This simplifies to:

a = -rac{3}{7}

So, we've found our first solution! a equals -3/7. This means that if we substitute -3/7 for a in the original equation, $(7a + 3)(4a - 12) = 0$, the equation will hold true. But remember, we had two possible equations to solve, so we're not done yet. We still need to solve the second equation, $4a - 12 = 0$, to find the other possible value of a. Let's move on to that now and see what we find!

Solving the Second Equation: $4a - 12 = 0$

Now, let's move on to the second equation we derived from the Zero Product Property: $4a - 12 = 0$. Just like with the first equation, our aim here is to isolate a on one side of the equation. We'll follow a similar process, using basic algebraic principles to manipulate the equation. First, we need to get rid of the constant term, which is -12 in this case. To do that, we'll add 12 to both sides of the equation. Remember, maintaining balance is key, so whatever we do to one side, we must do to the other. Adding 12 to both sides gives us:

$4a - 12 + 12 = 0 + 12$

This simplifies to:

$4a = 12$

Great! Now we have $4a$ on the left side, but we want a by itself. To isolate a, we need to eliminate the 4 that's multiplying it. We can do this by dividing both sides of the equation by 4. Again, we must perform the same operation on both sides to keep the equation balanced:

rac{4a}{4} = rac{12}{4}

This simplifies to:

$a = 3$

And there we have it! Our second solution is a equals 3. This means that if we substitute 3 for a in the original equation, $(7a + 3)(4a - 12) = 0$, the equation will also hold true. So, we've found both possible values for a that satisfy the equation. Now that we've solved both equations, let's take a moment to summarize our findings and make sure we've answered the question completely. We're in the home stretch now!

Summarizing the Solutions

Alright, let's take a step back and summarize what we've accomplished. We started with the equation $(7a + 3)(4a - 12) = 0$ and our mission was to find the values of a that make this equation true. We used the Zero Product Property to break down the equation into two simpler equations:

1. $7a + 3 = 0$
2. $4a - 12 = 0$

We then solved each of these equations separately. For the first equation, $7a + 3 = 0$, we found that:

a = -rac{3}{7}

For the second equation, $4a - 12 = 0$, we found that:

$a = 3$

So, we have two solutions for a: -3/7 and 3. This means that there are two values of a that will make the original equation $(7a + 3)(4a - 12) = 0$ true. We can write our solution set as:

a = \{-rac{3}{7}, 3\}

This set notation simply means that the possible values for a are -3/7 and 3. We've successfully found all the solutions to the equation! It's always a good idea to double-check your work, especially in math. You can do this by plugging each solution back into the original equation to make sure it holds true. If you substitute -3/7 for a in the original equation, you'll find that one of the factors becomes zero, making the entire product zero. Similarly, if you substitute 3 for a, you'll find that the other factor becomes zero, again making the entire product zero. This confirms that our solutions are correct. Great job, guys! You've tackled a potentially tricky problem by understanding and applying the Zero Product Property. Keep up the great work!