It's funny how sometimes the simplest-looking math problems can feel like a little puzzle, isn't it? Take the equation 3x² - 48 = 0. On the surface, it's just a few numbers and a variable. But diving into it reveals a neat little journey through the fundamentals of algebra.
Let's break it down, shall we? The goal here is to find the value, or values, of 'x' that make this equation true. Think of it like trying to find the secret key that unlocks the balance of the equation.
Our first step, as many of you might recall from your math classes, is to isolate the term with 'x' in it. So, we'll add 48 to both sides of the equation. This is like moving things around on a scale to keep it balanced.
3x² - 48 + 48 = 0 + 48
This simplifies to:
3x² = 48
Now, we want to get 'x²' all by itself. To do that, we divide both sides by 3.
3x² / 3 = 48 / 3
Which gives us:
x² = 16
Here's where it gets interesting. We're looking for a number that, when multiplied by itself, equals 16. This is where the concept of square roots comes into play. You might immediately think of 4, because 4 * 4 = 16. And you'd be absolutely right!
But here's a little twist that often catches people out: remember that a negative number multiplied by itself also results in a positive number. So, (-4) * (-4) also equals 16.
This means our equation has two solutions! We write this as:
x = ±4
So, the two values for 'x' that satisfy the equation 3x² - 48 = 0 are x = 4 and x = -4. It's a great example of how a single equation can hold multiple truths, and how understanding the properties of numbers, like square roots, is key to unlocking them. It’s a small victory, but a satisfying one, wouldn't you agree?
