Solving (a-7)(a-8) = 0: A Step-by-Step Guide
Hey guys! Let's dive into solving a classic quadratic equation presented in a slightly different form: (a-7)(a-8) = 0. This might look intimidating at first, but trust me, it's super manageable once you understand the core concept. This concept hinges on the Zero Product Property, which is your best friend when dealing with equations like these. So, let's break it down, make it easy, and get you feeling confident about tackling these types of problems.
Understanding the Zero Product Property
Before we jump into the specifics of our equation, let's make sure we're all on the same page about the Zero Product Property. This property is the key to unlocking solutions for equations where factors are multiplied together and set equal to zero. Simply put, the Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. That's it! It sounds simple, and it is simple, but it's incredibly powerful.
Think of it this way: If you have two numbers, let's call them x and y, and you know that x * y = 0, then you can confidently say that either x = 0, y = 0, or both x and y are zero. There's no other way to get a product of zero! This is the fundamental idea that we'll use to solve our equation. When we see an equation like (a-7)(a-8) = 0, we immediately recognize that we have two factors, (a-7) and (a-8), whose product is zero. The Zero Product Property tells us that one or both of these factors must be equal to zero. This understanding allows us to split the original equation into two simpler equations, which are much easier to solve. These simpler equations represent the two possible scenarios where the product is zero, and solving each of them will give us the solutions to the original equation. This is a common technique in algebra, so grasping this concept will be incredibly beneficial as you tackle more complex problems. We will use the Zero Product Property in the next section to solve our example.
Applying the Zero Product Property to (a-7)(a-8) = 0
Now that we've got a solid grasp of the Zero Product Property, let's put it to work on our equation: (a-7)(a-8) = 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 (a-7) and (a-8). So, we can set up two separate equations:
- a - 7 = 0
- a - 8 = 0
See how we've transformed one equation into two simpler ones? This is the magic of the Zero Product Property! Now, all we need to do is solve each of these equations for 'a'. The first equation, a - 7 = 0, is straightforward. To isolate 'a', we simply add 7 to both sides of the equation. This gives us a = 7. So, one possible solution is a = 7. This means that if we substitute 7 for 'a' in the original equation, the left-hand side will indeed equal zero. The second equation, a - 8 = 0, is solved in a similar way. We add 8 to both sides of the equation to isolate 'a'. This gives us a = 8. So, our second possible solution is a = 8. Again, if we substitute 8 for 'a' in the original equation, the left-hand side becomes zero. We've now found two values for 'a' that make the original equation true. We've effectively broken down the problem into manageable steps by applying the Zero Product Property, and we've successfully solved for 'a'. Remember, the key is to recognize the factored form and use the property to create simpler equations. This technique is crucial for solving many types of algebraic equations.
Solving the Equations: a - 7 = 0 and a - 8 = 0
Alright, let's knock out these two mini-equations we derived from the Zero Product Property. We've got:
- a - 7 = 0
- a - 8 = 0
These are both simple one-step equations, which makes our lives much easier. Let's tackle a - 7 = 0 first. Our goal here is to isolate 'a' on one side of the equation. To do this, we need to get rid of the '-7' that's hanging out with 'a'. The inverse operation of subtraction is addition, so we're going to add 7 to both sides of the equation. Remember, whatever you do to one side of an equation, you have to do to the other to keep things balanced! So, we add 7 to both sides:
a - 7 + 7 = 0 + 7
This simplifies to:
a = 7
Boom! We've found our first solution. a = 7 makes the equation a - 7 = 0 true. Now, let's move on to the second equation, a - 8 = 0. The process is exactly the same. We want to isolate 'a', and we have a '-8' to deal with. Again, we'll use the inverse operation of subtraction, which is addition. We'll add 8 to both sides of the equation:
a - 8 + 8 = 0 + 8
This simplifies to:
a = 8
Awesome! We've found our second solution. a = 8 makes the equation a - 8 = 0 true. So, by applying a simple addition to both sides, we've successfully solved these two equations. Remember, the key is to perform the same operation on both sides of the equation to maintain balance and isolate the variable you're trying to solve for. These skills are fundamental to algebra, and mastering them will make solving more complex equations much easier.
The Solutions: a = 7 and a = 8
Fantastic! We've done the heavy lifting and arrived at our solutions. After applying the Zero Product Property and solving the resulting equations, we found two values for 'a' that satisfy the original equation (a-7)(a-8) = 0. These solutions are:
- a = 7
- a = 8
That's it! We've successfully solved the equation. But, to be absolutely sure and to demonstrate a solid understanding, it's always a good idea to check our answers. We can do this by substituting each solution back into the original equation and verifying that it holds true. This process not only confirms our solutions but also reinforces our understanding of the equation and the solving process. Let's start by checking a = 7. Substitute 7 for 'a' in the original equation:
(7 - 7)(7 - 8) = 0
This simplifies to:
(0)(-1) = 0
Which is indeed true! So, a = 7 is definitely a solution. Now, let's check a = 8. Substitute 8 for 'a' in the original equation:
(8 - 7)(8 - 8) = 0
This simplifies to:
(1)(0) = 0
This is also true! So, a = 8 is also a valid solution. We've verified both solutions, giving us confidence that our work is correct. The solutions a = 7 and a = 8 are the values that make the original equation true. When solving equations, always remember the importance of checking your answers to ensure accuracy and deepen your understanding.
Conclusion: Mastering Quadratic Equations
Alright, guys, we've successfully navigated the equation (a-7)(a-8) = 0! We started by understanding the crucial Zero Product Property, which allowed us to break down a seemingly complex equation into simpler, manageable parts. We then applied this property to split the original equation into two linear equations: a - 7 = 0 and a - 8 = 0. Solving each of these equations was straightforward, involving a simple addition operation on both sides. This gave us our two solutions: a = 7 and a = 8.
To ensure our accuracy, we didn't just stop there. We took the extra step of verifying our solutions by substituting them back into the original equation. This crucial step confirmed that both a = 7 and a = 8 indeed satisfy the equation, giving us complete confidence in our answer. The process we've used here is a powerful one that extends far beyond this specific example. The Zero Product Property is a fundamental tool in solving quadratic equations, especially those presented in factored form. By mastering this property and the techniques we've discussed, you'll be well-equipped to tackle a wide range of algebraic problems.
Remember, the key to success in mathematics is practice. So, try applying these concepts to other similar equations. Look for equations where expressions are multiplied together and set equal to zero. Practice using the Zero Product Property to split them into simpler equations, and then solve for the variable. With consistent practice, you'll become more comfortable and confident in your problem-solving abilities. Keep up the great work, and you'll be conquering quadratic equations in no time! This approach is applicable not only to mathematics but to any problem-solving scenario: understand the core principle, break the problem into smaller parts, solve them individually, verify the solutions, and practice consistently.
