Solving Quadratic Equations Find X In 6x² - 30x - 84 = 0

Hey guys! Let's dive into solving a quadratic equation today. We've got a fun one: 6x² - 30x - 84 = 0. Our mission is to find the values of x that make this equation true. We'll break it down step by step, so don't worry if it looks intimidating at first.

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is basically an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants (numbers) and 'a' is not zero. The 'x' is our variable, and we're trying to find the values of 'x' that satisfy the equation. These values are also called roots or solutions of the equation. In our case, the equation 6x² - 30x - 84 = 0 perfectly fits this form, with a = 6, b = -30, and c = -84. Understanding this basic structure is the first step in tackling these equations, allowing us to identify the coefficients and the constant term, which are crucial for applying various solving methods. This general form helps us to recognize quadratic equations and apply appropriate techniques to find the solutions. Solving quadratic equations is a fundamental skill in algebra with wide applications in various fields, making it essential for students and professionals alike. Mastering this topic not only helps in academic pursuits but also in practical problem-solving scenarios, where quadratic relationships often appear. The coefficients and constants play a significant role in determining the shape and position of the parabola represented by the quadratic equation when graphed, further highlighting the importance of understanding these components.

Simplifying the Equation: A Key First Step

Now, looking at our equation 6x² - 30x - 84 = 0, you might notice that all the coefficients (6, -30, and -84) are divisible by 6. This is a fantastic opportunity! We can simplify the equation by dividing both sides by 6. This makes our lives much easier. So, let's do it: (6x² - 30x - 84) / 6 = 0 / 6. This simplifies to x² - 5x - 14 = 0. See? Much cleaner! Simplifying equations before diving into solving is a powerful technique. It reduces the size of the numbers we're dealing with, making calculations simpler and less prone to errors. This is especially helpful when dealing with larger coefficients. By dividing, we maintain the equation's balance while making it easier to handle. This step is not just about making the numbers smaller; it’s about setting ourselves up for success in the next steps of solving the equation. A simpler equation means simpler factoring or the application of the quadratic formula, ultimately leading to a more straightforward solution process. Moreover, this simplification highlights the importance of recognizing common factors in mathematical expressions, a skill that extends beyond quadratic equations and into various areas of algebra and beyond. The ability to simplify equations efficiently is a hallmark of strong mathematical proficiency, allowing for quicker and more accurate problem-solving.

Factoring: Our Go-To Method

Okay, we've got x² - 5x - 14 = 0. Now, we're going to try factoring. Factoring involves rewriting the quadratic expression as a product of two binomials. Think of it like reversing the FOIL (First, Outer, Inner, Last) method. We need to find two numbers that multiply to -14 (our 'c' term) and add up to -5 (our 'b' term). Let's think... What numbers fit the bill? After a bit of thought, -7 and 2 seem to work perfectly! -7 multiplied by 2 is -14, and -7 plus 2 is -5. Bingo! So, we can rewrite our equation as (x - 7)(x + 2) = 0. Factoring is a fundamental technique in algebra, and it's incredibly useful for solving quadratic equations. It’s like finding the puzzle pieces that fit together to form the original expression. The key to successful factoring lies in identifying the correct pair of numbers that satisfy both the multiplication and addition conditions. This often involves a bit of trial and error, but with practice, it becomes second nature. Factoring not only simplifies the solving process for quadratic equations but also provides a deeper understanding of the structure of algebraic expressions. It’s a skill that extends beyond equation solving, proving valuable in simplifying expressions, finding common factors, and more. Moreover, the ability to factor efficiently can significantly reduce the time and effort required to solve complex algebraic problems, making it an essential tool in any mathematician's toolkit.

Finding the Solutions: Setting Each Factor to Zero

We've factored our equation to (x - 7)(x + 2) = 0. Now comes the cool part! If the product of two things is zero, then at least one of them must be zero. This is a fundamental principle in mathematics. So, we can set each factor equal to zero: x - 7 = 0 and x + 2 = 0. Now we have two simple equations to solve. For x - 7 = 0, we add 7 to both sides to get x = 7. For x + 2 = 0, we subtract 2 from both sides to get x = -2. So, our solutions are x = 7 and x = -2. This step beautifully demonstrates the power of factoring in simplifying the process of finding solutions. By breaking down the quadratic equation into simpler, linear equations, we make the solutions easily accessible. The principle that if the product of two factors is zero, then at least one must be zero, is a cornerstone of algebra and is used extensively in solving various types of equations. It provides a direct path from factored form to individual solutions, making it an invaluable tool. Moreover, this step highlights the connection between the factors of a quadratic expression and its roots, reinforcing the understanding of algebraic structures. This method not only efficiently finds solutions but also enhances conceptual understanding, making it a crucial technique for anyone studying algebra.

The Solutions: Putting It All Together

We've done it! We found the solutions to the equation 6x² - 30x - 84 = 0. Our solutions are x = -2 and x = 7. The question asks us to insert the solutions from smallest to largest. So, the smaller value is x = -2, and the larger value is x = 7. Therefore:

menor x = -2 maior x = 7

We nailed it! Solving quadratic equations can seem daunting at first, but by breaking it down into manageable steps – simplifying, factoring, and setting factors to zero – it becomes much more approachable. Remember, practice makes perfect, so keep at it, and you'll become a quadratic equation-solving pro in no time! Understanding how to solve quadratic equations is a cornerstone of algebra, with applications extending far beyond the classroom. These equations model real-world phenomena in physics, engineering, economics, and many other fields, making the ability to solve them a valuable skill. From projectile motion to optimization problems, quadratic equations provide the framework for understanding and predicting outcomes. This reinforces the importance of mastering this algebraic skill for practical problem-solving. Moreover, the process of solving quadratic equations strengthens analytical thinking and problem-solving abilities, skills that are transferable to many different areas of life. Whether it's factoring, using the quadratic formula, or completing the square, each method provides a unique approach to tackling the problem, fostering adaptability and a deeper understanding of mathematical principles. With continued practice and a solid grasp of the fundamental concepts, anyone can confidently solve quadratic equations and apply this knowledge to solve real-world challenges.

Answer:

a) x₁ = -2 e x₂ = 7