Solve 2x² - 24x + 54 = 0: Step-by-Step Guide

Hey guys! Today, we're diving deep into solving a quadratic equation. Specifically, we're tackling the equation 2x² - 24x + 54 = 0. Quadratic equations might seem daunting at first, but trust me, with a few key techniques, they become super manageable. We'll break down the steps, explore different methods, and make sure you understand not just how to solve this particular equation, but also the general principles behind solving any quadratic equation. So, buckle up, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are all about. A quadratic equation is basically a polynomial equation of the second degree. That might sound a bit technical, but what it really means is that it has a term with x raised to the power of 2 (that's the "quadratic" part). The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers), and 'x' is the variable we're trying to find. The solutions to a quadratic equation are also called its roots.

Think of 'a', 'b', and 'c' as coefficients that determine the shape and position of a parabola when the equation is graphed. The roots, or solutions, are the points where the parabola intersects the x-axis. A quadratic equation can have two real roots, one real root (which we call a repeated root), or two complex roots. Understanding this foundational concept is crucial because it gives us a visual and conceptual understanding of what we're actually doing when we solve these equations. It's not just about crunching numbers; it's about finding where a curve crosses a line! Now that we've got the basics down, let's see how these principles apply to our specific problem, 2x² - 24x + 54 = 0.

Simplifying the Equation

Alright, let's get to grips with our equation: 2x² - 24x + 54 = 0. The first thing I always look for when solving quadratic equations (or any equation, really) is whether we can simplify it. Simplification makes the numbers easier to work with and reduces the chances of making mistakes. In this case, we can see that all the coefficients (2, -24, and 54) are divisible by 2. So, let's divide the entire equation by 2. This gives us a much cleaner equation to deal with: x² - 12x + 27 = 0. See how much nicer that looks?

Dividing by a common factor doesn't change the roots of the equation, because we're essentially just scaling the entire equation down. It's like looking at a smaller, but perfectly proportional, version of the same problem. This step is so important because it not only makes the arithmetic simpler, but it also sets us up for the next steps in solving the equation, such as factoring or using the quadratic formula. By reducing the coefficients, we make these processes less cumbersome and more straightforward. Now, with our simplified equation, x² - 12x + 27 = 0, we're ready to explore the different methods we can use to find the value of x. Let's move on and see how we can factor this beauty!

Method 1: Factoring the Quadratic Equation

Factoring is often the quickest and most elegant way to solve a quadratic equation, if it's factorable, of course. The idea behind factoring is to rewrite the quadratic expression as a product of two binomials. In our case, we want to express x² - 12x + 27 in the form (x + p)(x + q), where p and q are constants. To find p and q, we need to identify two numbers that multiply to give 27 (the constant term) and add up to -12 (the coefficient of the x term).

Think of it like a puzzle. We need two numbers that fit both criteria. Let's list the factors of 27: 1 and 27, 3 and 9. Since we need the numbers to add up to -12, we'll consider the negative factors as well: -1 and -27, -3 and -9. Bingo! -3 and -9 fit the bill perfectly: (-3) * (-9) = 27 and (-3) + (-9) = -12. So, we can rewrite our equation as (x - 3)(x - 9) = 0. Now, the magic happens. For the product of two factors to be zero, at least one of them must be zero. This gives us two possible solutions: x - 3 = 0 or x - 9 = 0. Solving these simple linear equations, we get x = 3 and x = 9. And there you have it! We've found the values of x by factoring. Factoring is a powerful technique, but it's not always easy to spot the factors. That's where our next method comes in handy.

Method 2: Using the Quadratic Formula

When factoring seems tricky or impossible, the quadratic formula is your best friend. It's a universal tool that works for any quadratic equation, no matter how messy the coefficients are. Remember the general form of a quadratic equation: ax² + bx + c = 0? The quadratic formula gives us the solutions for x in terms of a, b, and c:

x = [-b ± √(b² - 4ac)] / (2a)

Don't be intimidated by the formula! It looks a bit complex, but it's just a matter of plugging in the values and doing the arithmetic. In our simplified equation, x² - 12x + 27 = 0, we have a = 1, b = -12, and c = 27. Let's substitute these values into the formula:

x = [-(-12) ± √((-12)² - 4 * 1 * 27)] / (2 * 1)

Now, let's simplify step by step. First, we have -(-12), which is simply 12. Then, we calculate (-12)² which is 144, and 4 * 1 * 27 which is 108. So, inside the square root, we have 144 - 108 = 36. The square root of 36 is 6. Now our equation looks like this:

x = [12 ± 6] / 2

We have two possible solutions here, one with the plus sign and one with the minus sign. Let's calculate them separately.

For the plus sign: x = (12 + 6) / 2 = 18 / 2 = 9

For the minus sign: x = (12 - 6) / 2 = 6 / 2 = 3

Lo and behold! We get the same solutions as we did by factoring: x = 3 and x = 9. The quadratic formula might seem like a longer route, but it's a reliable method that always works. It's like having a Swiss Army knife in your math toolkit. Now, let's take a moment to reflect on what we've learned.

Verification and Conclusion

We've solved the quadratic equation 2x² - 24x + 54 = 0 using two different methods: factoring and the quadratic formula. Both methods gave us the same solutions: x = 3 and x = 9. But we're not done yet! It's always a good idea to verify our solutions to make sure we haven't made any mistakes along the way. To verify, we simply plug our solutions back into the original equation and see if they make it true.

Let's start with x = 3:

2(3)² - 24(3) + 54 = 2(9) - 72 + 54 = 18 - 72 + 54 = 0

It checks out! Now let's try x = 9:

2(9)² - 24(9) + 54 = 2(81) - 216 + 54 = 162 - 216 + 54 = 0

It checks out too! Both solutions are valid. So, we can confidently say that the solutions to the equation 2x² - 24x + 54 = 0 are x = 3 and x = 9. Solving quadratic equations is a fundamental skill in algebra, and we've covered two powerful methods today. Whether you prefer the elegance of factoring or the reliability of the quadratic formula, you now have the tools to tackle a wide range of quadratic equations. Remember, practice makes perfect! The more you work with these methods, the more comfortable and confident you'll become. Keep practicing, and you'll be solving quadratic equations like a pro in no time! This journey through solving 2x² - 24x + 54 = 0 underscores the importance of not just finding the answers, but understanding the process. By simplifying, choosing the right method (factoring or the quadratic formula), and verifying our solutions, we build a solid foundation in algebra. Now you're equipped to handle any quadratic equation that comes your way. Keep up the great work, and happy solving!