Solving 3(10x-7)(x^2+64) = 0: A Step-by-Step Guide
Hey guys! Today, we're diving into a fun math problem: solving the equation 3(10x-7)(x^2+64) = 0 for all possible values of x. This might look a little intimidating at first, but don't worry, we'll break it down step by step and make it super easy to understand. Think of it like a puzzle – we just need to find the right pieces to fit together.
Understanding the Basics: The Zero Product Property
Before we jump into the nitty-gritty, let's refresh a key concept called the Zero Product Property. This property is our best friend when solving equations like this. It basically says that if the product of two or more factors is zero, then at least one of those factors must be zero.
In simpler terms, if we have something like A * B = 0, then either A = 0, B = 0, or both A and B are zero. This is the golden rule we'll use to crack this equation.
Why is this important? Because our equation, 3(10x-7)(x^2+64) = 0, is already in a factored form. This means we have three factors multiplied together: 3, (10x-7), and (x^2+64). According to the Zero Product Property, at least one of these factors needs to be zero for the whole equation to be true. Let's investigate each factor one by one.
Factor 1: The Constant 3
The first factor is simply the number 3. Can 3 ever be equal to zero? Nope! 3 is always 3, so we can rule this factor out as a source of solutions. This is often the case with constant factors; they don't contain any variables, so they won't contribute to the solutions of the equation. It's like a red herring – it's there, but we don't need to worry about it.
Factor 2: The Linear Expression (10x - 7)
Now, let's look at the second factor: (10x - 7). This is a linear expression, meaning it involves x to the power of 1. To find the value of x that makes this factor zero, we set it equal to zero and solve for x:
10x - 7 = 0
Let's add 7 to both sides of the equation to isolate the term with x:
10x = 7
Now, we divide both sides by 10 to get x by itself:
x = 7/10
So, one solution to our equation is x = 7/10. This is a real number, and it's a perfectly valid solution. We've found our first piece of the puzzle! This means that when x is equal to 7/10, the entire expression 3(10x-7)(x^2+64) will equal zero. That's pretty cool, right?
Factor 3: The Quadratic Expression (x^2 + 64)
Finally, let's tackle the third factor: (x^2 + 64). This is a quadratic expression because it involves x squared. Again, we set this factor equal to zero and try to solve for x:
x^2 + 64 = 0
To isolate x squared, we subtract 64 from both sides:
x^2 = -64
Now, we need to take the square root of both sides to find x. But here's a twist: we're taking the square root of a negative number! Remember, the square of a real number is always non-negative (either positive or zero). There's no real number that, when squared, gives you -64. This is where imaginary numbers come into play.
We introduce the imaginary unit, denoted by i, which is defined as the square root of -1: i = √(-1). Using this, we can rewrite the square root of -64 as follows:
√(-64) = √(64 * -1) = √(64) * √(-1) = 8i
Since we're taking a square root, we need to consider both the positive and negative roots. So, we have two solutions from this factor:
x = 8i and x = -8i
These are complex solutions, meaning they involve the imaginary unit i. So, these are our last two pieces of the puzzle!
Putting It All Together: The Solutions
Alright, we've investigated each factor and found all the values of x that make the equation 3(10x-7)(x^2+64) = 0 true. Let's summarize our findings:
- From the factor (10x - 7), we found x = 7/10.
- From the factor (x^2 + 64), we found x = 8i and x = -8i.
Therefore, the solutions to the equation are x = 7/10, x = 8i, and x = -8i. We have one real solution and two complex solutions. Awesome!
Visualizing the Solutions (Optional)
While we can't easily visualize complex solutions on a standard number line (which represents real numbers), we can think about the real solution, x = 7/10, as a point on the number line. It's a little bit past the halfway point between 0 and 1. The complex solutions, 8i and -8i, exist on the imaginary axis, which is perpendicular to the real number line. If you delve deeper into complex numbers, you'll learn how to represent them graphically on a complex plane.
Key Takeaways
Let's recap the key steps we took to solve this equation:
- Understand the Zero Product Property: This is crucial for solving factored equations.
- Set each factor equal to zero: This allows us to find the potential solutions.
- Solve each resulting equation: This might involve linear equations, quadratic equations, or even the introduction of imaginary numbers.
- List all the solutions: Make sure you've accounted for all possible values of x.
By following these steps, you can tackle a wide range of equations with multiple factors. The more you practice, the more comfortable you'll become with these concepts. Math is like a muscle – the more you use it, the stronger it gets!
Practice Makes Perfect
Now that we've solved this equation together, why not try some similar problems on your own? Here are a few ideas to get you started:
- (x - 3)(x + 5)(2x - 1) = 0
- (x^2 - 9)(x + 2) = 0
- 4x(x^2 + 25) = 0
Remember, the key is to break the problem down into smaller steps and apply the Zero Product Property. Don't be afraid to make mistakes – that's how we learn! And if you get stuck, review the steps we've covered in this guide. You've got this!
Final Thoughts
So, there you have it! We've successfully solved the equation 3(10x-7)(x^2+64) = 0 and explored the world of real and complex solutions. Remember, math is all about understanding the underlying principles and applying them creatively. Keep practicing, keep exploring, and keep having fun with it! You're doing great, guys! Keep up the awesome work!