Solving 3x² - 6x + 1 = 0 Using The Quadratic Formula

Hey guys! Let's dive into solving a quadratic equation using the quadratic formula. It might sound intimidating, but trust me, it’s a super useful tool in your math arsenal. We're going to break down each step, so by the end of this, you’ll be a pro at solving these types of equations. Our mission today is to tackle the equation 3x² - 6x + 1 = 0. We will walk through each step, making sure you understand how to apply the quadratic formula effectively.

Understanding the Quadratic Formula

Before we jump into the problem, let’s refresh what the quadratic formula actually is. The quadratic formula is a way to solve equations that are in the form of ax² + bx + c = 0. You've probably seen it before, but here it is again for good measure:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

• a is the coefficient of the x² term
• b is the coefficient of the x term
• c is the constant term

This formula might look like a jumble of letters and symbols, but it’s a powerful tool once you get the hang of it. Essentially, it gives us the values of x that make the equation equal to zero. These values are also known as the roots or solutions of the quadratic equation. The key to using the quadratic formula effectively is correctly identifying a, b, and c from your equation and plugging them in accurately. Errors in substitution can lead to incorrect answers, so paying close attention to this step is crucial. Another thing to keep in mind is the ± symbol in the formula. This indicates that there are generally two solutions to a quadratic equation, one where you add the square root term and one where you subtract it. The term inside the square root, b² - 4ac, is called the discriminant, and it tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions; if it’s zero, we have one real solution (a repeated root); and if it’s negative, we have two complex solutions. Understanding the quadratic formula isn't just about memorizing it; it's about knowing when and how to apply it. It's a fundamental concept in algebra and is used extensively in various fields of mathematics and science. So, take your time, practice, and you'll master it in no time!

Step-by-Step Solution for 3x² - 6x + 1 = 0

Okay, now that we've got the quadratic formula fresh in our minds, let's apply it to our equation: 3x² - 6x + 1 = 0. The first thing we need to do is identify our a, b, and c values. Remember, a is the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term. In this equation:

• a = 3
• b = -6
• c = 1

Make sure you pay close attention to the signs! A negative sign can make a big difference in your answer. Now that we have our values, we're going to carefully plug them into the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). Substituting our values, we get:

x = [-(-6) ± √((-6)² - 4 * 3 * 1)] / (2 * 3)

Notice how we've replaced a, b, and c with their respective values. The next step is simplifying the expression. Let's start by simplifying the terms inside the square root and the other parts of the equation:

x = [6 ± √(36 - 12)] / 6

We've simplified -(-6) to 6, calculated (-6)² as 36, and multiplied 4 * 3 * 1 to get 12. Now, let's continue simplifying:

x = [6 ± √24] / 6

We've subtracted 12 from 36 under the square root. The next step is to simplify the square root. We can simplify √24 by finding its prime factors. 24 can be factored into 4 * 6, and 4 is a perfect square (2²). So, we can rewrite √24 as √(4 * 6), which simplifies to 2√6. Let's substitute that back into our equation:

x = [6 ± 2√6] / 6

Now we're almost there! We can simplify this fraction by dividing both terms in the numerator and the denominator by their greatest common divisor, which is 2:

x = (3 ± √6) / 3

And there you have it! We've solved the equation using the quadratic formula. Remember, the key to mastering this is practice. Try solving different quadratic equations, and you'll become more comfortable with the process.

Final Answer and Options

So, after grinding through the quadratic formula, we've landed on our solutions for x: x = (3 ± √6) / 3. Now, let’s match this up with the options given to us in the question. Looking at the options, we have:

A. x = (-3 ± 2√6) / 3 B. x = (3 ± 2√3) / 3 C. x = (-3 ± √6) / 3 D. x = (3 ± √6) / 3

Comparing our solution with the options, we can clearly see that option D matches our answer. Therefore, the correct answer is:

D. x = (3 ± √6) / 3

It’s awesome when everything lines up perfectly, isn't it? We correctly identified the coefficients, plugged them into the formula, simplified everything carefully, and arrived at the right answer. This process underscores the importance of precision and attention to detail in mathematics. Each step, from identifying a, b, and c to simplifying the square root, is crucial. A small error in any step can lead to a completely different solution. So, always double-check your work, especially when dealing with signs and square roots. Understanding how to solve quadratic equations using the quadratic formula is a fundamental skill in algebra. It's not just about finding the right answer; it's about understanding the process and being able to apply it to different problems. So, keep practicing, and you'll become more confident and proficient in solving these types of equations. Remember, math can be challenging, but it's also incredibly rewarding when you solve a problem successfully. So, celebrate your accomplishments and keep pushing forward!

Common Mistakes and How to Avoid Them

Alright, let's talk about some common oopsies people make when using the quadratic formula. Knowing these pitfalls can seriously level up your problem-solving game. One of the biggest mistakes is messing up the signs. Remember, the formula is x = [-b ± √(b² - 4ac)] / (2a). That “-b” part can be tricky if b itself is negative. For example, in our equation 3x² - 6x + 1 = 0, b is -6. So, -b becomes -(-6), which is positive 6. Always double-check those signs! Another common mistake is botching the calculation under the square root, which is the discriminant (b² - 4ac). Make sure you square b correctly and follow the order of operations (PEMDAS/BODMAS). It’s super easy to miscalculate this part, especially under pressure. Simplifying the square root can also be a stumbling block. In our problem, we had √24, which simplifies to 2√6. If you're not comfortable with simplifying radicals, practice factoring numbers into their prime factors. This skill is not just useful for the quadratic formula but for many other math problems too. Lastly, watch out for errors when simplifying the final fraction. We had (6 ± 2√6) / 6, which simplifies to (3 ± √6) / 3. Make sure you're dividing all terms in the numerator by the denominator. A good way to avoid these mistakes is to write out each step clearly and double-check your work as you go. It might take a little longer, but it’s way better than getting the wrong answer. Also, practice makes perfect! The more you use the quadratic formula, the more natural it will become, and the fewer mistakes you'll make. So, keep at it, guys! You’ve got this!

Tips for Mastering the Quadratic Formula

Okay, so you’re on your way to becoming a quadratic formula whiz! But let’s nail down some solid tips to really master this tool. First off, memorization is key. You gotta know that formula inside and out: x = [-b ± √(b² - 4ac)] / (2a). Write it down, say it out loud, make up a silly song about it – whatever works for you! The faster you can recall the formula, the smoother the problem-solving process will be. Next up, practice, practice, practice! Seriously, the more you use the quadratic formula, the more comfortable you'll become with it. Start with easier equations and gradually move on to more complex ones. You can find tons of practice problems online or in your textbook. Work through them step by step, and don’t be afraid to make mistakes – that’s how you learn! Another great tip is to check your answers. Once you've solved for x, plug your solutions back into the original equation to make sure they work. This is a fantastic way to catch any errors you might have made along the way. If your solutions don't make the equation true, you know you need to go back and check your work. Also, try to **_understand the