Solving Quadratic Equations: A Step-by-Step Guide

Introduction to Quadratic Equations

Let's dive into the world of quadratic equations! You might be thinking, "What exactly is a quadratic equation?" Well, it's essentially a polynomial equation of the second degree. This means the highest power of the variable (usually 'x' or in our case 'w') is 2. Quadratic equations pop up all over the place, from physics problems involving projectile motion to engineering calculations for bridge design. They're super useful, so understanding how to solve them is a valuable skill. 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 equation would become linear, not quadratic.

Before we jump into solving our specific equation, $5w^2 - 36w = 32$, it’s crucial to understand why we need specific methods for these types of equations. Linear equations (where the highest power of the variable is 1) can be solved by simple algebraic manipulation—isolating the variable on one side. However, the presence of the squared term in quadratic equations changes the game. We can't just isolate 'w' directly because it appears in both a squared term ($w^2$) and a linear term ($w$). This is where the fun begins! We'll explore how different techniques, like factoring, completing the square, and the quadratic formula, help us tackle this challenge. Think of these methods as tools in your mathematical toolbox, each suited for different situations. By mastering these tools, you'll be well-equipped to solve a wide range of quadratic equations. So, let's get started and unlock the secrets of these powerful equations!

Problem: 5w² - 36w = 32

Our specific problem is the quadratic equation 5w² - 36w = 32. Notice how it looks a little different from the standard form ax² + bx + c = 0. The first step in solving this equation is to rearrange it into the standard form. This involves moving all the terms to one side of the equation, leaving zero on the other side. In our case, we need to subtract 32 from both sides of the equation. This gives us: $5w^2 - 36w - 32 = 0$. Now, our equation is in the standard form, making it easier to apply various solving methods. Identifying 'a', 'b', and 'c' is the next crucial step. In our equation, 'a' is the coefficient of the $w^2$ term, which is 5. 'b' is the coefficient of the 'w' term, which is -36. And 'c' is the constant term, which is -32. These values are essential for both factoring and using the quadratic formula.

Understanding the coefficients is like understanding the ingredients in a recipe – you need to know what they are before you can start cooking! Correctly identifying 'a', 'b', and 'c' ensures that you'll plug the right values into the quadratic formula or use them correctly when factoring. This seemingly simple step is a common place where errors can occur, so double-checking these values is always a good idea. With our equation now in standard form and the coefficients identified, we’re ready to explore different methods to find the values of 'w' that satisfy the equation. These values are also known as the roots or solutions of the quadratic equation. There can be up to two real solutions for a quadratic equation, and finding them is our ultimate goal. So, let's delve into the techniques we can use to crack this mathematical puzzle!

Method 1: Factoring

Factoring is a powerful technique for solving quadratic equations, but it's not always straightforward. It relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if (x - p)(x - q) = 0, then either x - p = 0 or x - q = 0, which means x = p or x = q. This principle allows us to break down a quadratic equation into simpler linear equations. To factor the quadratic equation $5w^2 - 36w - 32 = 0$, we need to find two binomials that, when multiplied together, give us the original quadratic expression. This can sometimes be a bit of a trial-and-error process, especially when the coefficient of the squared term (our 'a' value) is not 1. One common strategy is to look for two numbers that multiply to give the product of 'a' and 'c' (in our case, 5 * -32 = -160) and add up to 'b' (-36).

Let's think about the factors of -160. We need a pair that has a difference of 36. After some mental math or perhaps a quick list of factors, we can find that -40 and 4 fit the bill: -40 * 4 = -160 and -40 + 4 = -36. Now, we can rewrite the middle term (-36w) using these two numbers: $5w^2 - 40w + 4w - 32 = 0$. Next, we factor by grouping. We group the first two terms and the last two terms: $(5w^2 - 40w) + (4w - 32) = 0$. Now, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out 5w, and from the second group, we can factor out 4: $5w(w - 8) + 4(w - 8) = 0$. Notice that we now have a common binomial factor, (w - 8). We can factor this out: $(5w + 4)(w - 8) = 0$. Now, we apply the principle we discussed earlier. Either 5w + 4 = 0 or w - 8 = 0. Solving these linear equations gives us our solutions. For 5w + 4 = 0, we subtract 4 from both sides and then divide by 5, giving us w = -4/5. For w - 8 = 0, we simply add 8 to both sides, giving us w = 8. So, our solutions are w = -4/5 and w = 8. Factoring is a neat way to solve quadratic equations, but it’s not always the easiest method, especially if the factors aren't obvious or if the equation has irrational solutions. In those cases, other methods might be more efficient.

Method 2: Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations. It works for any quadratic equation, regardless of whether it can be factored easily or not. This makes it a reliable and essential method to have in your mathematical toolkit. The quadratic formula is derived from the process of completing the square on the general form of the quadratic equation, $ax^2 + bx + c = 0$. The formula itself is: w = (-b ± √(b² - 4ac)) / 2a. Notice the ± symbol, which means there are potentially two solutions: one where we add the square root term and one where we subtract it. This corresponds to the two possible roots of a quadratic equation.

To apply the quadratic formula to our equation, $5w^2 - 36w - 32 = 0$, we first need to identify the values of 'a', 'b', and 'c', which we already did earlier: a = 5, b = -36, and c = -32. Now, we carefully substitute these values into the formula: w = (-(-36) ± √((-36)² - 4 * 5 * -32)) / (2 * 5). Let's break this down step by step. First, -(-36) simplifies to 36. Next, we calculate the discriminant, which is the term inside the square root: (-36)² - 4 * 5 * -32 = 1296 + 640 = 1936. So, our equation becomes: w = (36 ± √1936) / 10. Now, we need to find the square root of 1936. √1936 = 44. Therefore, w = (36 ± 44) / 10. This gives us two possible solutions. For the first solution, we add: w = (36 + 44) / 10 = 80 / 10 = 8. For the second solution, we subtract: w = (36 - 44) / 10 = -8 / 10 = -4/5. As you can see, we arrived at the same solutions we found using factoring: w = 8 and w = -4/5. The quadratic formula might seem a bit intimidating at first, but with practice, it becomes a straightforward method. It's particularly useful when factoring is difficult or impossible. The discriminant (b² - 4ac) also provides valuable information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it's zero, there is exactly one real solution (a repeated root). And if it's negative, there are two complex solutions. Understanding the discriminant can give you a sneak peek at the type of solutions you should expect before you even finish solving the equation.

Conclusion

We've successfully solved the quadratic equation $5w^2 - 36w = 32$ using two different methods: factoring and the quadratic formula. Both methods led us to the same solutions: w = 8 and w = -4/5. This demonstrates the power and versatility of these techniques. Factoring is a great option when the quadratic expression can be easily factored, as it often provides a quicker solution. However, the quadratic formula is a more general method that works for all quadratic equations, making it an indispensable tool in your mathematical arsenal. Understanding both methods gives you the flexibility to choose the most efficient approach for a given problem.

Solving quadratic equations is a fundamental skill in algebra and has applications in various fields, including physics, engineering, and computer science. Mastering these techniques will not only help you in your math courses but also provide you with valuable problem-solving skills that can be applied in many real-world situations. Remember, practice makes perfect! The more you work with quadratic equations, the more comfortable and confident you'll become in solving them. Don't be afraid to try different methods and see which one works best for you. And remember, mathematics is a journey of discovery, so enjoy the process of learning and exploring!

For further learning and practice on quadratic equations, you can visit resources like Khan Academy's Quadratic Equations Section. This website provides comprehensive lessons, practice exercises, and videos to help you deepen your understanding of quadratic equations and other algebraic concepts. Happy solving!