Solve (x+7)(x-9)=25 By Completing The Square

Hey guys! Today, we're diving into a fun math problem: solving the equation (x+7)(x-9)=25 using the completing the square method. This might sound a bit intimidating, but trust me, we'll break it down step-by-step so it's super easy to follow. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the question is asking. We have a quadratic equation, which means it has an $x^2$ term. Our mission is to find the values of $x$ that make this equation true. The completing the square method is a nifty technique to rewrite the equation in a form that's easier to solve. It's all about transforming our equation into a perfect square trinomial. But what exactly is a perfect square trinomial? Well, it's a trinomial (an expression with three terms) that can be factored into the square of a binomial (an expression with two terms). For example, $x^2 + 6x + 9$ is a perfect square trinomial because it can be factored into $(x+3)^2$. See? The completing the square method leverages this concept to make solving quadratic equations a breeze. So, let's gear up to tackle our main equation, (x+7)(x-9)=25, and see how this method works its magic!

Step-by-Step Solution

Okay, let's get to the fun part – actually solving the equation! We'll break it down into manageable steps so you can follow along easily. First things first, we need to expand the left side of the equation: (x+7)(x-9)=25. This means we're going to multiply the two binomials together. Remember the good ol' FOIL method? (First, Outer, Inner, Last). Applying FOIL gives us: $x^2 - 9x + 7x - 63 = 25$. Now, let's simplify by combining like terms: $x^2 - 2x - 63 = 25$. Next up, we want to get all the terms on one side of the equation, so we'll subtract 25 from both sides: $x^2 - 2x - 63 - 25 = 0$. This simplifies to: $x^2 - 2x - 88 = 0$. Awesome! We've got our equation in the standard quadratic form: $ax^2 + bx + c = 0$. Now, the real magic of completing the square begins. We need to transform the left side into a perfect square trinomial. To do this, we'll focus on the first two terms, $x^2 - 2x$. We take half of the coefficient of our $x$ term (which is -2), square it, and add it to both sides of the equation. Half of -2 is -1, and (-1) squared is 1. So, we add 1 to both sides: $x^2 - 2x + 1 - 88 = 1$. Now we can rewrite the left side as a perfect square: $(x - 1)^2 - 88 = 1$. Almost there! Let's isolate the squared term by adding 88 to both sides: $(x - 1)^2 = 89$. Now we're ready for the final steps to solve for $x$.

Completing the Square Method

Alright, we've reached a crucial point in our journey of solving the equation (x+7)(x-9)=25 by completing the square. We've successfully transformed the equation to $(x - 1)^2 = 89$. Now, it's time to unleash the power of square roots! To get rid of the square on the left side, we'll take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots. This gives us: $x - 1 = pms 89$. Now, we have two possible equations: $x - 1 = 90 89$ and $x - 1 = - 91 89$. Let's solve for $x$ in each case. For the first equation, $x - 1 = 92 89$, we simply add 1 to both sides: $x = 1 + 93 89$. And for the second equation, $x - 1 = - 94 89$, we also add 1 to both sides: $x = 1 - 95 89$. Voila! We've found our solutions. So, the values of $x$ that satisfy the equation are $x = 1 + 96 89$ and $x = 1 - 97 89$. This means our final answer is $x = 1 98

99 89$. See? Completing the square might have seemed like a daunting task at first, but by breaking it down into smaller steps, we've conquered it together!

Identifying the Correct Answer

So, we've done the hard work and found that the solutions for $x$ are $x = 1 + 100 89$ and $x = 1 - 101 89$. Now, let's take a look at the answer choices provided and see which one matches our solution. We have: A. $x=1 102

103 89$ B. $x=1 104

105 87$ C. $x=-2$ or 14 D. $x=-4$ or 6 Comparing our solution, $x = 1 106

107 89$, with the answer choices, we can clearly see that option A, $x=1 108

109 89$, is the correct one! We've successfully navigated through the steps of completing the square and pinpointed the right answer. Give yourselves a pat on the back, guys! You've tackled a quadratic equation like champs. This process highlights how important it is to not only solve the problem but also to double-check your solution against the given options to ensure accuracy. Now, let's recap the key steps we took to get here, just to solidify our understanding.

Key Takeaways and Recap

Okay, guys, let's take a moment to recap what we've learned today about solving the equation (x+7)(x-9)=25 by completing the square. This method is a powerful tool for tackling quadratic equations, and here are the key steps we followed: 1. Expand and Simplify: We started by expanding the left side of the equation using the FOIL method and then simplified it to get the equation in the standard quadratic form, $ax^2 + bx + c = 0$. 2. Isolate Terms: We moved all the terms to one side of the equation, setting it equal to zero. 3. Complete the Square: This is the heart of the method! We took half of the coefficient of the $x$ term, squared it, and added it to both sides of the equation. This transformed the left side into a perfect square trinomial. 4. Factor and Simplify: We factored the perfect square trinomial and simplified the equation. 5. Take the Square Root: We took the square root of both sides, remembering to consider both positive and negative roots. 6. Solve for x: Finally, we solved for $x$ in each of the resulting equations. Remember, the goal of completing the square is to rewrite the quadratic equation in a form where we can easily isolate $x$. By understanding these steps and practicing them, you'll become a pro at using this method. And remember, math is like building with LEGOs – each step builds upon the previous one. So, make sure you're solid on each step before moving on. Now, let's think about why this method is so useful and where else you might encounter it.

Why Completing the Square Matters

You might be wondering, why bother learning completing the square? Well, guys, it's not just about solving this one equation. This method is a fundamental concept in algebra and has several important applications. Firstly, completing the square helps us understand the structure of quadratic equations. It allows us to rewrite any quadratic equation in vertex form, which gives us valuable information about the parabola's vertex (the highest or lowest point) and axis of symmetry. This is super useful when graphing quadratic functions. Secondly, completing the square is the foundation for deriving the quadratic formula, which is a universal tool for solving any quadratic equation. So, mastering this method gives you a deeper understanding of where the quadratic formula comes from. Thirdly, the technique of completing the square extends beyond quadratics. It's a useful strategy in calculus, particularly when dealing with integrals involving quadratic expressions. You might also encounter it in other areas of math and science where quadratic relationships pop up. Think about physics problems involving projectile motion or optimization problems in economics. Understanding completing the square gives you a versatile tool in your problem-solving arsenal. So, while it might seem like a specific technique for a specific type of equation, it's actually a gateway to understanding broader mathematical concepts and applications. Keep practicing, and you'll see how this method becomes second nature!

Practice Problems and Further Learning

Alright, now that we've conquered the equation (x+7)(x-9)=25 using the completing the square method, it's time to put your skills to the test! Practice makes perfect, so let's talk about some ways you can solidify your understanding and become even more confident in solving quadratic equations. First off, try tackling similar problems. Look for equations in the form $(x + a)(x + b) = c$ and challenge yourself to solve them using the same steps we covered today. You can also find plenty of practice problems in textbooks, online resources, and worksheets. Don't be afraid to start with easier problems and gradually work your way up to more complex ones. Another great way to learn is by explaining the method to someone else. Teaching is a fantastic way to reinforce your own understanding. Try explaining the steps of completing the square to a friend or family member. If you're looking for additional resources, there are tons of videos and articles online that delve deeper into the topic. Khan Academy is a fantastic resource for math tutorials, and you can also find helpful explanations on websites like Mathway and Purplemath. Remember, the key is consistent practice and a willingness to learn from your mistakes. Math isn't always easy, but with persistence and the right strategies, you can master any concept. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this, guys!