Solving 3x² - 12x = 9 By Completing The Square A Step-by-Step Guide

Completing the square is a crucial technique in algebra for solving quadratic equations, and it's super useful for putting quadratics into vertex form. Let's break down how to complete the square for the equation $3x^2 - 12x = 9$. We'll go through each step meticulously, so you can nail this method every time. Stick around, and by the end, you’ll be a pro at completing the square!

Understanding the Basics of Completing the Square

Before we dive into the specifics of our equation, let's quickly recap the general idea behind completing the square. The goal is to transform a quadratic equation from the standard form $ax^2 + bx + c = 0$ into the vertex form $a(x - h)^2 + k = 0$. The vertex form makes it super easy to identify the vertex $(h, k)$ of the parabola, which is incredibly helpful for graphing and solving optimization problems. So, why is this such a powerful tool? Well, it gives us a straightforward way to find the solutions (or roots) of the quadratic equation and reveals the turning point of the parabola, providing deep insights into the behavior of the quadratic function. When we complete the square, we're essentially creating a perfect square trinomial, which can be factored into the square of a binomial. This process involves a bit of algebraic manipulation, but once you grasp the concept, it becomes second nature. We are aiming to rewrite the quadratic expression in a form that allows us to easily take the square root, which simplifies the process of solving for x. The method is rooted in the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$. By recognizing and creating these patterns, we can transform any quadratic equation into a solvable form. Whether you’re tackling tough algebra problems or prepping for calculus, mastering this technique is a game-changer. So, let's jump into the steps and make sure you're ready to complete the square like a champ!

Step-by-Step Solution for 3x² - 12x = 9

Now, let's tackle the equation $3x^2 - 12x = 9$ step-by-step. Trust me, following along carefully will make it crystal clear! The first key step in completing the square when you have a leading coefficient (a number in front of the $x^2$ term) is to make sure that coefficient is 1. Why? Because the standard completing the square formulas work best when the quadratic is in the form $x^2 + bx + c$. So, to do this, we need to divide every term in the equation by the leading coefficient. In our case, the leading coefficient is 3. Dividing both sides of the equation by 3 gives us: $x^2 - 4x = 3$. See? Much simpler already! Now that we have the equation in this form, the next step is to focus on completing the perfect square trinomial on the left side. Remember, a perfect square trinomial is a trinomial that can be factored into $(x + n)^2$ or $(x - n)^2$. To figure out what constant term we need to add to both sides, we take half of the coefficient of our x term (which is -4), square it, and add the result to both sides. Half of -4 is -2, and squaring -2 gives us 4. So, we add 4 to both sides of the equation: $x^2 - 4x + 4 = 3 + 4$. This simplifies to $x^2 - 4x + 4 = 7$. The left side is now a perfect square trinomial! We can factor it as $(x - 2)^2$. So, our equation becomes $(x - 2)^2 = 7$. And there you have it! We've completed the square. This form of the equation is incredibly useful because it directly tells us about the vertex of the parabola represented by the quadratic equation. In this case, we can see that the vertex has an x-coordinate of 2. The equation also gives us a clear path to solving for x if we need to find the roots. By taking the square root of both sides, we can isolate x and find the solutions. This step-by-step approach ensures that you’re not just memorizing a process but truly understanding each manipulation. So, keep practicing, and you’ll become a master at completing the square!

Identifying the Correct Answer

Okay, guys, we’ve walked through the steps of completing the square for $3x^2 - 12x = 9$. Now, let’s pinpoint which of the given options is the correct one. Remember, after dividing by 3 and completing the square, we arrived at the equation $(x - 2)^2 = 7$. So, let’s review the options:

• A. $(x - 2)^2 = 7$
• B. $(x + 2)^2 = 7$
• C. $(3x - 2)^2 = 13$
• D. $(3x - 2)^2 = 20$

Comparing our result with the options, it’s clear that option A, $(x - 2)^2 = 7$, perfectly matches what we found. Options B, C, and D don't align with our step-by-step solution. Option B has the wrong sign inside the parenthesis, and options C and D have a different form altogether, as they didn’t correctly divide the original equation by 3 before completing the square. The key takeaway here is the importance of following each step precisely. Each manipulation we performed – dividing by the leading coefficient, finding the correct constant to add, and factoring the perfect square trinomial – was crucial in arriving at the correct answer. So, when you're tackling these problems, make sure to take your time, double-check each step, and keep the core principles of completing the square in mind. By doing so, you can confidently identify the correct answer and boost your algebra skills!

Common Mistakes to Avoid

Alright, let's chat about some common pitfalls people stumble into when completing the square. Knowing these mistakes can seriously level up your accuracy and save you from frustration. One of the biggest slip-ups is forgetting to divide the entire equation by the leading coefficient before you start completing the square. Remember, the goal is to get the equation into the form $x^2 + bx + c = d$ before you find the constant to add. If you skip this step, you'll end up with a completely wrong equation. Another frequent error is messing up the sign when finding the constant to add. You need to take half of the coefficient of the x term and then square it. For example, if your x term is -4x, you take half of -4, which is -2, and then square -2 to get 4. Make sure you're squaring the entire result, including the negative sign, because a negative number squared is always positive. Another spot where people often trip up is only adding the constant to one side of the equation. Remember, the golden rule of algebra is to keep the equation balanced. Whatever you do to one side, you must do to the other. So, if you add 4 to complete the square on the left side, you must also add 4 to the right side. Failing to do so will throw off your entire solution. Lastly, make sure you correctly factor the perfect square trinomial. Once you've added the correct constant, the left side of your equation should be easily factorable into a binomial squared. For example, $x^2 - 4x + 4$ factors to $(x - 2)^2$. Double-check your factoring to make sure you have the correct binomial. By keeping these common mistakes in mind and practicing consistently, you'll be able to complete the square with confidence and avoid those pesky errors. Remember, it's all about attention to detail and understanding the fundamental steps.

Practice Problems and Further Learning

To truly master completing the square, you need to put in the reps! Practice makes perfect, as they say. So, let’s talk about some practice problems and resources to help you become a completing-the-square whiz. First off, try working through various quadratic equations on your own. Start with simpler equations and gradually move on to more complex ones. For instance, try completing the square for $x^2 + 6x = 16$ or $2x^2 - 8x = 10$. Remember to follow each step carefully: divide by the leading coefficient if necessary, find the constant to add, add it to both sides, factor the perfect square trinomial, and simplify. The more you practice, the more natural the process will feel. Another fantastic way to sharpen your skills is to use online resources. Websites like Khan Academy, Paul’s Online Math Notes, and others offer tons of practice problems, video tutorials, and step-by-step explanations. Khan Academy, in particular, has excellent modules on completing the square, complete with practice exercises and progress tracking. These resources are super helpful because they let you check your work and identify any areas where you might be struggling. Don't hesitate to use these tools to their fullest extent. Textbooks are also a great resource. Look for sections on quadratic equations and completing the square in your algebra textbook. Work through the examples provided and try the practice problems at the end of the section. Many textbooks also have answer keys, so you can check your solutions. If you’re still finding the topic challenging, consider working with a tutor or joining a study group. Sometimes, having someone explain the concept in a different way or working through problems collaboratively can make a big difference. Tutoring services, both in-person and online, can provide personalized instruction and help you tackle specific difficulties. And don't underestimate the power of just talking through the problems with a friend or classmate. Explaining the steps out loud can solidify your understanding and help you spot any errors. So, get out there, grab some problems, and start practicing. With consistent effort and the right resources, you’ll be completing the square like a pro in no time!

Conclusion

Alright, guys, we’ve covered a lot of ground! Completing the square is a fundamental technique in algebra, and now you've got the tools to tackle it head-on. We started by understanding the basics, then walked through a step-by-step solution for $3x^2 - 12x = 9$. We identified the correct answer, discussed common mistakes to avoid, and explored practice problems and further learning resources. The key takeaway here is that completing the square isn't just about memorizing a process; it's about understanding each step and why it works. By mastering this technique, you’ll be able to solve quadratic equations more effectively, put them into vertex form, and gain a deeper understanding of quadratic functions. Whether you’re acing your algebra class, prepping for standardized tests, or just expanding your math skills, completing the square is a valuable tool in your arsenal. So, keep practicing, stay patient, and remember that every mistake is a learning opportunity. With consistent effort, you'll become more confident and proficient. And hey, if you ever get stuck, don’t hesitate to revisit this guide or seek out additional resources. You’ve got this! Now, go out there and conquer those quadratic equations!