Solving X² - 10x + 25 = 35 A Comprehensive Guide

Hey guys! Let's dive into solving a quadratic equation today. We've got a fun one: x² - 10x + 25 = 35. Our mission, should we choose to accept it, is to find the value(s) of x that make this equation true. We'll break it down step-by-step, making sure everyone can follow along. So, buckle up and let’s get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are all about. A quadratic equation is essentially a polynomial equation of the second degree. That fancy way of saying it just means it has a term with x squared (x²) as the highest power of x. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. Now, why are these equations so important? Well, they pop up everywhere in real-world applications, from physics and engineering to economics and computer science. Think about projectile motion, the design of bridges, or even modeling financial markets – quadratic equations are often lurking behind the scenes. There are several methods we can use to solve them, such as factoring, completing the square, and using the quadratic formula. Each method has its own strengths and is suitable for different types of equations. Factoring is great when the equation can be easily broken down into simpler expressions. Completing the square is a powerful technique that can be used for any quadratic equation. And the quadratic formula? It's the trusty old workhorse that always gets the job done, no matter how messy the equation looks. So, with this basic understanding in our toolkit, let's tackle the equation at hand and see which method will work best for us.

Method 1: Simplifying and Factoring

When faced with a quadratic equation, one of the first things we should try is simplifying it. Our equation is x² - 10x + 25 = 35. The goal here is to manipulate the equation so that one side equals zero. This is crucial because it allows us to use some powerful techniques, like factoring, to find the solutions. So, let's get to it! We'll start by subtracting 35 from both sides of the equation. This gives us x² - 10x + 25 - 35 = 0, which simplifies to x² - 10x - 10 = 0. Now, take a good look at the left side of the equation. Does it look familiar? Sometimes, we can factor the quadratic expression into two binomials. Factoring is like reverse multiplication – we're trying to find two expressions that, when multiplied together, give us our original quadratic. In this case, we're looking for two numbers that multiply to -10 and add up to -10. Hmmm, this might be a bit tricky. It doesn't seem like there are any nice, whole numbers that fit the bill. When simple factoring doesn't work, it's a sign that we might need to use a different method, such as completing the square or the quadratic formula. But don't worry, we're not giving up yet! We've just learned that this particular equation might need a more sophisticated approach. Factoring is a fantastic tool when it works, but it's not the only trick in our toolbox. So, let's move on to another method and see if we can crack this equation open.

Method 2: Completing the Square

Alright, guys, let's try a different approach: completing the square. This method might sound a little intimidating at first, but trust me, it's a powerful technique for solving quadratic equations. Completing the square involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. For example, x² + 6x + 9 is a perfect square trinomial because it can be factored as (x + 3)². So, how do we make our equation, x² - 10x - 10 = 0, look like that? First, let's isolate the terms with x on one side of the equation. We can do this by adding 10 to both sides, which gives us x² - 10x = 10. Now comes the clever part. We need to add a number to both sides of the equation that will make the left side a perfect square trinomial. To find this number, we take half of the coefficient of our x term (which is -10), square it, and add it to both sides. Half of -10 is -5, and (-5)² is 25. So, we add 25 to both sides: x² - 10x + 25 = 10 + 25. This simplifies to x² - 10x + 25 = 35. Hooray! The left side is now a perfect square trinomial. We can factor it as (x - 5)². So our equation becomes (x - 5)² = 35. Now we're in business! We can easily solve for x by taking the square root of both sides. This method is super useful because it turns a tricky quadratic equation into something much more manageable. Let's keep going and see how this unfolds!

Continuing with Completing the Square

Now that we've got our equation in the form (x - 5)² = 35, let's finish solving for x using the completing the square method. The next step is to 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 is super important because quadratic equations often have two solutions! So, taking the square root of both sides gives us √(x - 5)² = ±√35. This simplifies to x - 5 = ±√35. Now, all we need to do is isolate x. We can do this by adding 5 to both sides of the equation. This gives us x = 5 ± √35. And there we have it! We've found our solutions for x. This means that there are two values of x that will satisfy the original equation: x = 5 + √35 and x = 5 - √35. Completing the square can be a bit of a journey, but it's a fantastic way to solve quadratic equations. It's especially helpful when the equation doesn't factor easily. Plus, it gives us a solid understanding of the structure of quadratic equations. So, we've successfully navigated this method and found our answers. But just for kicks, let's explore another method to solve this equation – the quadratic formula. It's like having a backup plan, and it's always good to have options, right?

Method 3: Using the Quadratic Formula

Okay, folks, let's pull out the big guns: the quadratic formula! This formula is a lifesaver for solving any quadratic equation, no matter how messy it looks. It's like the Swiss Army knife of quadratic equations – always reliable and ready to go. Remember the general form of a quadratic equation: ax² + bx + c = 0. The quadratic formula tells us that the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a). It might look a bit intimidating at first glance, but trust me, it's not as scary as it seems. It's just a matter of plugging in the right values. So, let's identify a, b, and c in our equation. We started with x² - 10x + 25 = 35, but we rewrote it as x² - 10x - 10 = 0. Now it's clear that a = 1, b = -10, and c = -10. Let's plug these values into the quadratic formula: x = (-(-10) ± √((-10)² - 4 * 1 * -10)) / (2 * 1). Now it's just a matter of simplifying. First, we have -(-10), which is just 10. Then, inside the square root, we have (-10)², which is 100. We also have -4 * 1 * -10, which is +40. So, our expression becomes: x = (10 ± √(100 + 40)) / 2. This simplifies to x = (10 ± √140) / 2. Now, we can simplify the square root. √140 can be written as √(4 * 35), which is 2√35. So, our equation becomes x = (10 ± 2√35) / 2. Finally, we can divide both terms in the numerator by 2: x = 5 ± √35. And guess what? We got the same answer as when we completed the square! The quadratic formula is a powerful tool, and it's always good to have it in your arsenal. It's especially useful when factoring is difficult or completing the square seems too cumbersome. So, we've conquered this equation using yet another method. We're on a roll!

Final Answer and Conclusion

Alright, we've tackled the equation x² - 10x + 25 = 35 using two different methods: completing the square and the quadratic formula. And guess what? We arrived at the same solution both times! This gives us extra confidence that we've nailed it. Our solutions for x are x = 5 + √35 and x = 5 - √35. We can write this more compactly as x = 5 ± √35. So, if we look back at the options provided, the correct answer is A. x = 5 ± √35. Solving quadratic equations might seem like a daunting task at first, but with the right tools and techniques, it becomes much more manageable. We've seen how factoring can be a quick and easy method when it works, but when things get a bit trickier, completing the square and the quadratic formula are there to save the day. The important thing is to understand the underlying concepts and practice applying these methods. The more you practice, the more comfortable you'll become with quadratic equations. And remember, math is like any other skill – the more you use it, the better you get at it. So, keep practicing, keep exploring, and keep having fun with math! You guys are awesome, and I know you can conquer any equation that comes your way.

Therefore, the final answer is:

A. x = 5 ± √35