Solving X² + 10x + 25 = 0 Quadratic Equation Step-by-Step
Hey guys! Today, we're diving deep into the fascinating world of quadratic equations. Specifically, we're going to tackle the equation x² + 10x + 25 = 0. Don't worry if that looks a bit intimidating – we'll break it down step by step, making sure you understand exactly how to solve it. Quadratic equations are fundamental in mathematics and have applications in various fields like physics, engineering, and even computer science. Mastering them is crucial for anyone looking to excel in these areas. So, grab your pencils and paper, and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That might sound like a mouthful, but it simply means it's an equation where the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the x² term would disappear, and we'd be left with a linear equation instead.
In our equation, x² + 10x + 25 = 0, we can easily identify the coefficients: a = 1, b = 10, and c = 25. Understanding these coefficients is the first step in choosing the right method to solve the equation. There are several ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its own strengths and weaknesses, and the best approach often depends on the specific equation we're dealing with. For example, some equations are easily factored, while others require the quadratic formula. Recognizing which method is most efficient will save you time and effort in the long run. Think of it like having different tools in a toolbox – each tool is best suited for a particular job. Similarly, each method for solving quadratic equations is most effective in certain situations.
Factoring the Equation
Okay, let's get back to our equation: x² + 10x + 25 = 0. The first method we'll try is factoring. Factoring involves rewriting the quadratic expression as a product of two binomials. If we can find two binomials that multiply together to give us our original quadratic expression, we can then set each binomial equal to zero and solve for x. This works because if the product of two factors is zero, then at least one of the factors must be zero. This is a fundamental principle in algebra and is the basis for solving many types of equations, not just quadratic ones. It’s like saying if A times B equals zero, then either A is zero, or B is zero, or both are zero.
To factor x² + 10x + 25, we need to find two numbers that add up to 10 (the coefficient of the x term) and multiply to 25 (the constant term). Think about it for a moment… what two numbers fit that description? If you guessed 5 and 5, you're absolutely right! 5 + 5 = 10, and 5 * 5 = 25. This means we can rewrite our quadratic expression as (x + 5)(x + 5). Notice how the numbers we found (5 and 5) appear in the binomials. This is the key to factoring – finding the numbers that satisfy both the addition and multiplication conditions. Now, our equation looks like this: (x + 5)(x + 5) = 0. Or, we can even write it as (x + 5)² = 0. This is a crucial step because it transforms our quadratic equation into a form that’s easy to solve.
Solving for x
Now that we've factored our equation to (x + 5)² = 0, the next step is super straightforward. To solve for x, we simply need to set the factor (x + 5) equal to zero. Why? Because if (x + 5)² equals zero, then (x + 5) must also equal zero. It's like saying if a square is zero, then the side of the square must also be zero. So, we have the equation x + 5 = 0. To isolate x, we subtract 5 from both sides of the equation. This gives us x = -5. And that's it! We've found the solution to our quadratic equation.
But wait, there's a little more to this solution than meets the eye. Notice that we have (x + 5) appearing twice in our factored equation. This means that x = -5 is a repeated root, or a root with a multiplicity of 2. What does that mean in simple terms? It means that the quadratic equation has only one distinct solution, but that solution occurs twice. In graphical terms, this means that the parabola represented by the equation touches the x-axis at only one point, x = -5. It doesn’t cross the x-axis, it just touches it and bounces back. Understanding the concept of repeated roots is important for a deeper understanding of quadratic equations and their graphs.
Alternative Methods: The Quadratic Formula
While factoring worked perfectly well for this particular equation, it's not always the most efficient method. Some quadratic equations are difficult or even impossible to factor using simple techniques. That's where the quadratic formula comes in handy. The quadratic formula is a universal solution for any quadratic equation of the form ax² + bx + c = 0. It's a powerful tool that will always give you the correct solution(s), regardless of whether the equation is factorable or not. Think of it as your trusty Swiss Army knife for solving quadratic equations – it can handle any situation!
The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. Don't let the symbols intimidate you – it's just a formula that you plug in the coefficients a, b, and c from your equation. Let's apply it to our equation, x² + 10x + 25 = 0, where a = 1, b = 10, and c = 25. Plugging these values into the formula, we get:
x = (-10 ± √(10² - 4 * 1 * 25)) / (2 * 1)
Now, let's simplify step by step. First, we calculate the expression inside the square root:
10² - 4 * 1 * 25 = 100 - 100 = 0
So, our formula becomes:
x = (-10 ± √0) / 2
Since the square root of 0 is 0, we have:
x = -10 / 2 = -5
As you can see, the quadratic formula also gives us the solution x = -5. Notice that we only get one solution in this case because the discriminant (the expression inside the square root, b² - 4ac) is zero. When the discriminant is zero, the quadratic equation has exactly one real solution, which is a repeated root. This confirms our earlier finding using the factoring method. The quadratic formula might seem more complex than factoring for this specific equation, but it's a valuable method to know for situations where factoring isn't straightforward.
Alternative Methods: Completing the Square
Another powerful method for solving quadratic equations is completing the square. This method is particularly useful because it can be used to derive the quadratic formula itself! Completing the square involves manipulating the quadratic equation into a form where one side is a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + k)² or (x - k)², where k is a constant. By completing the square, we can rewrite the equation in a way that allows us to easily solve for x. It's like transforming a puzzle into a shape that's easier to handle and solve.
Let's apply completing the square to our equation, x² + 10x + 25 = 0. Notice that the left side of the equation is already a perfect square trinomial! We recognized this earlier when we factored the equation. It factors neatly into (x + 5)². However, let's go through the steps of completing the square as if we didn't already know this, just to illustrate the process. The general steps for completing the square are as follows:
- Move the constant term to the right side of the equation. In our case, we already have 0 on the right side, so we don't need to do this step.
- Take half of the coefficient of the x term (b), square it, and add it to both sides of the equation. The coefficient of our x term is 10. Half of 10 is 5, and 5 squared is 25. Notice that 25 is already present on the left side of our equation, which is why it's a perfect square trinomial.
- Factor the left side as a perfect square trinomial. As we know, x² + 10x + 25 factors into (x + 5)².
So, our equation becomes (x + 5)² = 0. From here, we proceed as we did before: take the square root of both sides (which gives us x + 5 = 0) and solve for x, which gives us x = -5. Completing the square, in this case, is a bit like using a fancy tool to do a simple job, since the equation was already in a perfect form for this method. However, the process illustrates the general technique, which is crucial for solving more complex quadratic equations that aren't easily factorable.
Conclusion
So, guys, we've successfully solved the quadratic equation x² + 10x + 25 = 0 using three different methods: factoring, the quadratic formula, and completing the square. We found that the solution is x = -5, and it's a repeated root. This means the parabola represented by the equation touches the x-axis at only one point. Understanding these different methods is essential for tackling various types of quadratic equations and for building a strong foundation in algebra. Remember, each method has its strengths, and the best approach depends on the specific equation you're working with. Keep practicing, and you'll become a quadratic equation-solving pro in no time! This journey into solving quadratic equations highlights not just the mathematical techniques, but also the strategic thinking involved in choosing the most efficient method. Just like a chef selects the right ingredients and tools for a recipe, mathematicians choose the best methods to solve problems. By mastering these techniques, you're not just learning to solve equations; you're developing a problem-solving mindset that will serve you well in many areas of life. Keep exploring, keep questioning, and keep solving!
