Solving 6x^2 + 11x - 10 = 0 A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of quadratic equations and tackling a specific problem: 6x² + 11x - 10 = 0. If you've ever felt a little intimidated by these equations, don't worry – we're going to break it down step-by-step, making it super clear and easy to understand. Whether you're a student prepping for an exam, a math enthusiast looking to sharpen your skills, or just curious about how these equations work, you're in the right place. We'll explore different methods to solve this equation, discuss the underlying concepts, and provide plenty of tips to help you master this topic. So, grab your pencils and notebooks, and let's get started on this mathematical adventure together!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree. This means the highest power of the variable (usually '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 0. These equations pop up in various fields, from physics and engineering to economics and computer science, making them a fundamental concept in mathematics.

Think about it – the curve of a projectile's path, the shape of a satellite dish, or even the calculations behind financial models often involve quadratic equations. The solutions to these equations, also known as roots or zeros, represent the points where the parabola (the graph of a quadratic equation) intersects the x-axis. Understanding how to find these solutions is crucial for many real-world applications. Plus, mastering quadratic equations opens the door to more advanced mathematical concepts, so it's definitely a skill worth developing. Now that we've got a handle on the basics, let's move on to the fun part: solving our equation!

Methods to Solve Quadratic Equations

When it comes to solving quadratic equations, we have several powerful tools at our disposal. Each method has its own strengths and is suitable for different types of equations. We'll be focusing on three primary methods: factoring, using the quadratic formula, and completing the square. For our equation, 6x² + 11x - 10 = 0, we'll primarily use factoring and the quadratic formula, as they are often the most straightforward approaches for this type of problem. However, we'll also briefly touch upon completing the square to give you a complete picture.

  • Factoring: This method involves breaking down the quadratic expression into two binomial factors. When the product of these factors equals zero, at least one of the factors must be zero, allowing us to find the solutions. It's like finding the building blocks that make up the equation.
  • Quadratic Formula: This is a universal method that works for any quadratic equation. The formula is x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients from the standard form of the equation. Think of it as a reliable Swiss Army knife for solving quadratic equations.
  • Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. While it can be a bit more involved, it provides a deep understanding of the structure of quadratic equations and is particularly useful in certain situations, such as deriving the quadratic formula itself.

We're going to dive deep into factoring and the quadratic formula to solve our equation, so get ready to put these methods into action!

Solving 6x² + 11x - 10 = 0 by Factoring

Let's kick things off by tackling our equation using the factoring method. Factoring involves breaking down the quadratic expression into two binomials. This method relies on our ability to recognize patterns and manipulate the equation into a form that makes it easier to solve. For the equation 6x² + 11x - 10 = 0, the goal is to find two binomials that, when multiplied together, give us the original quadratic expression.

Here's how we can approach this: First, we look for two numbers that multiply to give us the product of the leading coefficient (6) and the constant term (-10), which is -60. At the same time, these two numbers must add up to the middle coefficient, which is 11. After some thought, we find that the numbers 15 and -4 fit the bill perfectly (15 * -4 = -60 and 15 + (-4) = 11). Next, we rewrite the middle term (11x) using these two numbers: 6x² + 15x - 4x - 10 = 0. Now, we factor by grouping. We group the first two terms and the last two terms: (6x² + 15x) + (-4x - 10) = 0. From the first group, we can factor out 3x, and from the second group, we can factor out -2: 3x(2x + 5) - 2(2x + 5) = 0. Notice that we now have a common factor of (2x + 5). We factor this out: (2x + 5)(3x - 2) = 0. Finally, we set each factor equal to zero and solve for x: 2x + 5 = 0 and 3x - 2 = 0. Solving these gives us x = -5/2 and x = 2/3. So, the solutions to our equation are x = -5/2 and x = 2/3. Awesome, right? Factoring can be a bit like solving a puzzle, but once you get the hang of it, it's a powerful tool for solving quadratic equations. Let's move on to the next method to see another way to crack this equation!

Applying the Quadratic Formula to 6x² + 11x - 10 = 0

Now, let's tackle the same equation, 6x² + 11x - 10 = 0, using the quadratic formula. This method is like having a magic key that unlocks the solutions to any quadratic equation, no matter how complex. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a, where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0.

In our case, we have a = 6, b = 11, and c = -10. Let's plug these values into the formula and see what happens. First, we calculate the discriminant, which is the part under the square root: b² - 4ac = 11² - 4 * 6 * (-10) = 121 + 240 = 361. The square root of 361 is 19, so we're in good shape. Now, we substitute everything into the formula: x = (-11 ± 19) / (2 * 6). This gives us two possible solutions: x = (-11 + 19) / 12 = 8 / 12 = 2/3 and x = (-11 - 19) / 12 = -30 / 12 = -5/2. Voila! We've arrived at the same solutions as we did with factoring: x = 2/3 and x = -5/2. The quadratic formula is a reliable method because it works every time, regardless of whether the equation is easily factorable. It might seem a bit intimidating at first, but with practice, it becomes second nature. Plus, it's a fantastic tool to have in your math arsenal. Now that we've conquered this equation using two different methods, let's take a step back and appreciate what we've learned!

Comparison of Methods and When to Use Them

We've successfully solved the quadratic equation 6x² + 11x - 10 = 0 using both factoring and the quadratic formula. Now, let's take a moment to compare these methods and discuss when it's best to use each one. Each method has its strengths and weaknesses, and knowing when to apply each can save you time and effort. It's like having different tools in a toolbox – each is designed for a specific task.

  • Factoring: Factoring is often the quickest method when the quadratic equation can be easily factored. It's like finding the perfect key to a lock – it's fast and efficient. However, not all quadratic equations are easily factorable, and attempting to factor a complex equation can sometimes be time-consuming. Factoring is best suited for equations where the coefficients are relatively small and the roots are rational numbers. If you spot a clear way to factor the equation, go for it! It's often the most elegant solution.
  • Quadratic Formula: The quadratic formula is the universal tool for solving quadratic equations. It works every time, regardless of the complexity of the equation or the nature of the roots. It's like having a master key that opens any lock. While it might involve a bit more calculation than factoring, it's a foolproof method. The quadratic formula is particularly useful when the equation is not easily factorable, or when you need to find solutions that are irrational or complex numbers. When in doubt, the quadratic formula is your best friend. It ensures you'll always find the solutions, even if they're not immediately obvious.

In the case of 6x² + 11x - 10 = 0, both methods worked well, but factoring required a bit more insight to find the right combination of factors. The quadratic formula, on the other hand, was a straightforward application of a formula. Ultimately, the choice of method depends on the specific equation and your personal preference. The more you practice, the better you'll become at recognizing which method is the most efficient for each situation. So keep practicing, guys! You're doing great.

Real-World Applications of Quadratic Equations

Alright, so we've become pretty good at solving quadratic equations, but you might be wondering,