Solve 5u² - 13u = 6: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of quadratic equations. Specifically, we're going to tackle the equation 5u² - 13u = 6. Don't worry, it might look intimidating at first, but we'll break it down step-by-step so you can conquer these problems with confidence. Whether you're a student prepping for an exam or just brushing up on your math skills, this guide is for you. We'll cover everything from the initial setup to the final solution, making sure you understand the why behind each step, not just the how. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are and why they're important. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, u) 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 0. These equations pop up all over the place in math and science, from calculating the trajectory of a projectile to optimizing areas and volumes. Think about it: any time you're dealing with curves or parabolic shapes, you're likely to encounter a quadratic equation. They're the workhorses behind many real-world applications, making them a crucial concept to master. Now, why do we need to solve them? Solving a quadratic equation means finding the values of the variable that make the equation true. These values are also known as the roots or solutions of the equation. For our equation, 5u² - 13u = 6, finding the solutions will give us the u values that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation you're dealing with. In this guide, we'll focus on one of the most common and versatile methods: factoring.

Step 1: Setting the Stage - Transforming the Equation

Alright, let's get our hands dirty with the actual problem. The first thing we need to do when solving a quadratic equation is to get it into the standard form: ax² + bx + c = 0. Our original equation is 5u² - 13u = 6. Notice that it's almost in the standard form, but we have that pesky '6' on the right side of the equation. To move it over, we need to subtract 6 from both sides. Remember, whatever you do to one side of the equation, you must do to the other to keep things balanced. This gives us: 5u² - 13u - 6 = 0. Now, we're talking! This form is crucial because it sets us up perfectly for the next step: factoring. By rearranging the equation into this standard form, we've made it much easier to identify the coefficients a, b, and c. In our case, a = 5, b = -13, and c = -6. These values will be important as we move forward with the factoring process. Think of this step as preparing your ingredients before you start cooking. You wouldn't start baking a cake without measuring out your flour and sugar, right? Similarly, we need to transform our equation into the standard form before we can apply the factoring method. This ensures we're working with a format that's conducive to finding the solutions.

Step 2: The Art of Factoring - Decomposing the Quadratic

Now comes the fun part: factoring! Factoring a quadratic equation involves breaking it down into two binomial expressions that, when multiplied together, give you the original quadratic. This might sound tricky, but with a bit of practice, you'll get the hang of it. Our equation is 5u² - 13u - 6 = 0. To factor this, we need to find two numbers that multiply to give us the product of a and c (which is 5 * -6 = -30) and add up to b (which is -13). This is a crucial step, so let's take our time. Think of it like solving a puzzle – we need to find the right pieces that fit together. Let's list out some factors of -30: 1 and -30, -1 and 30, 2 and -15, -2 and 15, 3 and -10, -3 and 10, 5 and -6, -5 and 6. Looking at these pairs, we can see that 2 and -15 fit the bill perfectly. They multiply to -30 and add up to -13. Now, we rewrite the middle term (-13u) using these two numbers: 5u² + 2u - 15u - 6 = 0. Notice that we've simply split the -13u into 2u and -15u. This doesn't change the equation, but it allows us to factor by grouping. Next, we group the terms in pairs: (5u² + 2u) + (-15u - 6) = 0. Now, we factor out the greatest common factor (GCF) from each pair. From the first pair, we can factor out u, and from the second pair, we can factor out -3: u(5u + 2) - 3(5u + 2) = 0. Notice something cool? Both terms now have a common factor of (5u + 2). This is a good sign – it means we're on the right track! Finally, we factor out the (5u + 2) term: (5u + 2)(u - 3) = 0. And there you have it! We've successfully factored the quadratic equation. This step is like the backbone of solving quadratic equations by factoring. Once you've mastered this, the rest is smooth sailing.

Step 3: The Zero Product Property - Unlocking the Solutions

We've factored our quadratic equation into (5u + 2)(u - 3) = 0. Now, it's time to use a powerful tool called the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if AB = 0, then either A = 0 or B = 0 (or both). This is the key to unlocking our solutions. Applying this property to our factored equation, we get two separate equations: 5u + 2 = 0 and u - 3 = 0. Each of these equations is much simpler to solve than the original quadratic. They're just linear equations! Let's solve the first one: 5u + 2 = 0. To isolate u, we first subtract 2 from both sides: 5u = -2. Then, we divide both sides by 5: u = -2/5. So, one solution is u = -2/5. Now, let's tackle the second equation: u - 3 = 0. To isolate u, we simply add 3 to both sides: u = 3. So, our second solution is u = 3. The Zero Product Property is like a magic key that transforms a factored quadratic equation into two solvable linear equations. It's a fundamental concept in algebra, and understanding it is crucial for solving not just quadratic equations, but many other types of equations as well.

Step 4: The Grand Finale - Presenting the Solutions

We've done the hard work, guys! We've factored the equation, applied the Zero Product Property, and solved for u. Now, it's time to present our solutions in a clear and concise manner. We found two solutions for the equation 5u² - 13u = 6: u = -2/5 and u = 3. These are the values of u that make the original equation true. To double-check our work, we can plug these values back into the original equation and see if they satisfy it. This is always a good practice to ensure we haven't made any mistakes along the way. Let's start with u = -2/5: 5(-2/5)² - 13(-2/5) = 6 5(4/25) + 26/5 = 6 4/5 + 26/5 = 6 30/5 = 6 6 = 6 It checks out! Now, let's try u = 3: 5(3)² - 13(3) = 6 5(9) - 39 = 6 45 - 39 = 6 6 = 6 It checks out as well! So, we can confidently say that our solutions are correct. When presenting your solutions, it's always a good idea to clearly state them, perhaps even highlight them. This makes it easy for anyone reading your work to see the final answer. In this case, we can say: "The solutions to the equation 5u² - 13u = 6 are u = -2/5 and u = 3." Presenting the solutions clearly is like putting the final touches on a masterpiece. It's the culmination of all your hard work, and it's important to showcase it effectively.

Conclusion: Mastering Quadratic Equations

Woo-hoo! We did it! We successfully solved the quadratic equation 5u² - 13u = 6 by factoring. We started by transforming the equation into standard form, then we factored it into two binomial expressions. We used the Zero Product Property to split the equation into two linear equations, solved each one, and finally, presented our solutions. Solving quadratic equations might seem daunting at first, but as you can see, it's a manageable process when you break it down into steps. The key is to practice regularly and understand the underlying concepts. Factoring is just one method for solving quadratic equations. There are other methods, such as completing the square and using the quadratic formula, which you can explore to expand your problem-solving toolkit. Each method has its advantages and disadvantages, and the best approach often depends on the specific equation you're dealing with. But mastering factoring is a fantastic starting point. It's a versatile technique that can be applied to many different types of quadratic equations. Remember, math is like building a house. You need a strong foundation to build upon. Understanding the basics, like factoring, is crucial for tackling more complex problems later on. So, keep practicing, keep exploring, and never stop learning! And hey, if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available, including teachers, tutors, and online communities. You've got this, guys! Keep up the great work!