Solve X² + 6x - 7 = 0 Quadratic Equation Step-by-Step
Introduction to Quadratic Equations
Hey guys! Let's dive into the world of quadratic equations. If you've ever stumbled upon an equation that looks like ax² + bx + c = 0, you've met a quadratic equation. These equations are super important in math and have tons of real-world applications, from figuring out the trajectory of a ball to designing bridges. In this article, we're going to break down how to solve one specific quadratic equation: x² + 6x - 7 = 0. We'll explore different methods, making sure you understand each step along the way. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable x is 2. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are also known as its roots or zeros, which represent the x-intercepts of the quadratic function's graph. These roots can be real or complex numbers, and a quadratic equation has exactly two roots, although they may be the same value (repeated roots). Quadratic equations appear in numerous fields, such as physics (projectile motion), engineering (structural design), economics (modeling growth), and computer science (optimization algorithms). Understanding how to solve them is a fundamental skill in mathematics and a powerful tool for solving problems across various domains. So, buckle up, and let's get started on this quadratic adventure!
Why This Equation Matters
This particular equation, x² + 6x - 7 = 0, is a classic example that helps illustrate key concepts in solving quadratics. It's got a straightforward structure, making it perfect for understanding the different methods we can use. We'll tackle it using factoring, which is a neat trick when it works, and then we'll double-check our answers with the quadratic formula, just to be thorough. Understanding how to solve x² + 6x - 7 = 0 is a stepping stone to solving more complex quadratic equations. This equation is carefully chosen because it's factorable, which allows us to demonstrate a simple and elegant solution method. Factoring involves breaking down the quadratic expression into a product of two binomials. This technique not only provides the solutions quickly but also deepens our understanding of the relationship between the coefficients and the roots of the equation. Moreover, this equation can also be solved using other methods like completing the square or the quadratic formula, providing an opportunity to compare and contrast different approaches. By mastering this equation, you'll gain confidence in your ability to tackle a wide range of quadratic problems and develop a strong foundation for more advanced mathematical concepts. The insights gained from solving x² + 6x - 7 = 0 extend far beyond this specific example, equipping you with the skills to analyze and solve quadratic equations in various mathematical and real-world contexts. So, let's dive in and explore the fascinating world of quadratic equations, one step at a time!
Overview of Methods
There are several ways to crack a quadratic equation. We'll focus on two main methods here: factoring and the quadratic formula. Factoring is like a puzzle – we try to rewrite the equation as a product of two expressions. The quadratic formula is a universal tool that works every time, no matter how messy the equation looks. Knowing both methods gives you flexibility and helps you choose the best approach for different problems. Factoring is an efficient method when the quadratic expression can be easily decomposed into binomial factors, making it a quick way to find the roots. However, not all quadratic equations can be factored easily, which is where the quadratic formula comes in handy. The quadratic formula is a powerful tool that provides a guaranteed solution for any quadratic equation, regardless of its complexity or the nature of its roots. Understanding both methods not only enhances your problem-solving skills but also gives you a deeper insight into the structure and properties of quadratic equations. Each method offers a unique perspective on the equation, helping you to appreciate the versatility and elegance of mathematical tools. By mastering these methods, you'll be well-equipped to tackle any quadratic equation that comes your way and to apply these techniques in more advanced mathematical contexts. So, let's get started and explore these methods in detail, so you can confidently solve quadratic equations like a pro!
Method 1: Factoring x² + 6x - 7 = 0
Factoring is a slick way to solve quadratic equations when it's possible. The idea is to rewrite our equation in a form that makes the solutions pop out. For x² + 6x - 7 = 0, we're looking for two numbers that multiply to -7 and add up to 6. Let’s break it down step by step. Factoring is a powerful technique for solving quadratic equations because it transforms the problem into a simpler one. Instead of dealing with a quadratic expression, we break it down into a product of two linear expressions. This method leverages the fact that if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to find the roots of the quadratic equation by setting each factor equal to zero and solving for x. Factoring not only provides a straightforward way to solve quadratic equations but also enhances our understanding of the structure and behavior of quadratic expressions. It helps us to see the connection between the coefficients of the quadratic equation and its roots, making it a valuable tool in our mathematical arsenal. By mastering factoring, we can efficiently solve a wide range of quadratic equations and gain a deeper appreciation for the elegance and power of algebraic manipulations. So, let's dive in and explore the steps involved in factoring the equation x² + 6x - 7 = 0, and unlock the secrets hidden within this quadratic expression!
Step 1: Find the Right Numbers
Think about factors of -7. We've got 1 and -7, or -1 and 7. Which pair adds up to 6? Bingo! It's -1 and 7. This is the heart of factoring – finding the right combination of numbers. This step is crucial because it lays the foundation for rewriting the quadratic expression in a factorable form. The key is to identify two numbers that satisfy two conditions simultaneously: their product must equal the constant term (c) of the quadratic equation, and their sum must equal the coefficient of the linear term (b). In the case of x² + 6x - 7 = 0, we need two numbers that multiply to -7 and add up to 6. This requires a careful consideration of both the magnitude and the sign of the factors. By systematically exploring the possible factor pairs of -7, we can identify the pair that meets both conditions. The numbers -1 and 7 fit the bill perfectly, as (-1) * (7) = -7 and (-1) + 7 = 6. This critical step transforms the quadratic expression into a form that can be easily factored, setting the stage for finding the solutions of the equation. So, take your time, think through the possibilities, and you'll master this essential factoring technique in no time!
Step 2: Rewrite and Factor
Now we rewrite our equation as (x - 1)(x + 7) = 0. See how -1 and 7 pop up? That's no accident! This step is where the magic happens. We've turned a quadratic equation into a product of two simpler expressions. Rewriting the quadratic expression as a product of two binomial factors is the essence of the factoring method. The numbers we found in the previous step, -1 and 7, play a crucial role here. These numbers are used to construct the binomial factors, which are expressions of the form (x + p) and (x + q), where p and q are the numbers we identified. In our case, the binomial factors are (x - 1) and (x + 7). When we multiply these factors, we get back the original quadratic expression: (x - 1)(x + 7) = x² + 7x - x - 7 = x² + 6x - 7. This confirms that our factoring is correct. The equation (x - 1)(x + 7) = 0 is now in a form that makes it easy to find the solutions. This is because if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for x, leading us to the roots of the quadratic equation. So, by rewriting and factoring, we've transformed a seemingly complex problem into a simple one, setting the stage for finding the solutions effortlessly!
Step 3: Find the Solutions
To make (x - 1)(x + 7) = 0 true, either (x - 1) = 0 or (x + 7) = 0. Solving these gives us x = 1 and x = -7. These are the solutions to our equation! Finding the solutions is the final step in the factoring process, where we leverage the factored form of the quadratic equation to determine the values of x that satisfy the equation. The principle we use here is the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, the factored form of the equation is (x - 1)(x + 7) = 0. This means that either (x - 1) = 0 or (x + 7) = 0. We solve each of these linear equations separately to find the solutions for x. Setting (x - 1) = 0, we add 1 to both sides and get x = 1. Similarly, setting (x + 7) = 0, we subtract 7 from both sides and get x = -7. Therefore, the solutions to the quadratic equation x² + 6x - 7 = 0 are x = 1 and x = -7. These values are also known as the roots or zeros of the quadratic equation, and they represent the x-intercepts of the corresponding quadratic function's graph. By finding these solutions, we've successfully solved the quadratic equation using the factoring method, demonstrating the power and elegance of this technique. So, congratulations on mastering factoring – you're one step closer to becoming a quadratic equation whiz!
Method 2: Using the Quadratic Formula
Okay, so factoring is cool, but it doesn't always work. That's where the quadratic formula swoops in to save the day! This formula can solve any quadratic equation, no matter how complicated. Let's see how it works for x² + 6x - 7 = 0. The quadratic formula is a universal tool for solving quadratic equations, providing a guaranteed solution for any equation in the form ax² + bx + c = 0. Unlike factoring, which relies on the ability to rewrite the quadratic expression as a product of binomial factors, the quadratic formula works regardless of whether the equation is factorable or not. This makes it an indispensable method for solving quadratic equations that cannot be easily factored. The quadratic formula is derived from the process of completing the square, and it expresses the solutions x in terms of the coefficients a, b, and c of the quadratic equation. By plugging in the values of a, b, and c into the formula, we can directly calculate the roots of the equation, whether they are real or complex numbers. The quadratic formula is not only a powerful problem-solving tool but also a testament to the beauty and elegance of mathematical formulas. It encapsulates the solutions of all quadratic equations in a single, concise expression, highlighting the power of mathematical abstraction and generalization. So, let's dive in and explore how to apply the quadratic formula to solve the equation x² + 6x - 7 = 0, and discover the magic of this remarkable formula!
The Formula
The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a. It looks a bit scary, but trust me, it's your friend! Let's break it down. The quadratic formula is a cornerstone of algebra, providing a direct method for finding the solutions of any quadratic equation in the form ax² + bx + c = 0. This formula expresses the roots x in terms of the coefficients a, b, and c, allowing us to calculate the solutions without having to factor the equation or complete the square. The formula is given by x = (-b ± √(b² - 4ac)) / 2a. The ± symbol indicates that there are two possible solutions, one with the plus sign and one with the minus sign. The expression inside the square root, b² - 4ac, is called the discriminant, and it plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root (a repeated root); and if it is negative, the equation has two complex roots. The quadratic formula is not just a formula; it's a powerful tool that encapsulates the solutions of all quadratic equations in a single expression. It's a testament to the beauty and elegance of mathematics, highlighting the power of abstraction and generalization. By mastering the quadratic formula, you'll be equipped to solve any quadratic equation that comes your way, regardless of its complexity or the nature of its roots. So, let's embrace this formula and unlock its potential to solve a wide range of quadratic problems!
Step 1: Identify a, b, and c
In our equation, x² + 6x - 7 = 0, a = 1, b = 6, and c = -7. This is like labeling the ingredients for our formula recipe. Identifying the coefficients a, b, and c is the first crucial step in applying the quadratic formula to solve a quadratic equation. These coefficients are the numerical values that multiply the terms x², x, and the constant term in the equation ax² + bx + c = 0. In our example equation, x² + 6x - 7 = 0, we can easily identify the coefficients by comparing it to the general form. The coefficient a is the number multiplying x², which in this case is 1 (since x² is the same as 1x²). The coefficient b is the number multiplying x, which is 6. And the coefficient c is the constant term, which is -7. Correctly identifying a, b, and c is essential because these values are directly plugged into the quadratic formula to calculate the solutions. A mistake in this step can lead to incorrect solutions, so it's important to be careful and double-check your work. Once you've identified the coefficients, you're ready to plug them into the formula and crank out the solutions. So, let's make sure we have our ingredients right before we start cooking up the solutions to this quadratic equation!
Step 2: Plug and Chug
Substitute these values into the formula: x = (-6 ± √(6² - 4 * 1 * -7)) / (2 * 1). Now it's just about doing the math carefully. Substituting the values of a, b, and c into the quadratic formula is a crucial step that transforms the general formula into a specific equation that we can solve. Once we've identified the coefficients, we replace the variables a, b, and c in the formula with their corresponding numerical values. In our example, we have a = 1, b = 6, and c = -7. Plugging these values into the formula x = (-b ± √(b² - 4ac)) / 2a, we get x = (-6 ± √(6² - 4 * 1 * -7)) / (2 * 1). This step is where the abstract formula becomes a concrete equation that we can manipulate to find the solutions. The key is to be careful and accurate when substituting the values, paying attention to the signs and order of operations. A small mistake in this step can lead to significant errors in the final solutions. After substituting the values, the next step is to simplify the expression and perform the necessary calculations. This involves evaluating the square root, performing the arithmetic operations, and finally, obtaining the solutions for x. So, let's plug in those values, roll up our sleeves, and get ready to simplify and solve this equation!
Step 3: Simplify
Let's simplify: x = (-6 ± √(36 + 28)) / 2 becomes x = (-6 ± √64) / 2, which simplifies to x = (-6 ± 8) / 2. We're almost there! Simplifying the expression after substituting the values into the quadratic formula is a critical step in obtaining the solutions. This involves performing the arithmetic operations in the correct order, following the rules of precedence (PEMDAS/BODMAS). In our example, after substituting the values, we have x = (-6 ± √(6² - 4 * 1 * -7)) / (2 * 1). The first step in simplifying is to evaluate the expression inside the square root: 6² - 4 * 1 * -7 = 36 + 28 = 64. So, the equation becomes x = (-6 ± √64) / 2. Next, we evaluate the square root: √64 = 8. This simplifies the equation to x = (-6 ± 8) / 2. Now, we have two separate equations to solve, one with the plus sign and one with the minus sign. This is where we split the ± symbol into two cases. Simplifying the expression step by step ensures that we arrive at the correct solutions without making any arithmetic errors. It's important to be meticulous and double-check each step to maintain accuracy. The goal of simplifying is to reduce the expression to its simplest form, making it easier to calculate the final solutions for x. So, let's keep simplifying, one step at a time, until we reach the final answers!
Step 4: Find the Solutions (Again!)
So, x = (-6 + 8) / 2 = 1 and x = (-6 - 8) / 2 = -7. Hey, look familiar? We got the same answers as before! Finding the solutions is the culmination of all our efforts in applying the quadratic formula. After simplifying the expression, we arrive at two separate equations, one with the plus sign and one with the minus sign. These equations represent the two possible values of x that satisfy the quadratic equation. In our example, we have x = (-6 + 8) / 2 and x = (-6 - 8) / 2. Solving the first equation, x = (-6 + 8) / 2, we get x = 2 / 2 = 1. Solving the second equation, x = (-6 - 8) / 2, we get x = -14 / 2 = -7. Therefore, the solutions to the quadratic equation x² + 6x - 7 = 0 are x = 1 and x = -7. These are the same solutions we obtained earlier using the factoring method, which confirms the correctness of our calculations. Finding the solutions is not just about getting the right answers; it's also about understanding the meaning of these values. The solutions represent the roots or zeros of the quadratic equation, which are the x-intercepts of the corresponding quadratic function's graph. By finding these points, we gain a deeper insight into the behavior of the quadratic function and its relationship to the x-axis. So, congratulations on solving the equation using the quadratic formula – you've added another powerful tool to your mathematical toolkit!
Conclusion
There you have it! We've solved x² + 6x - 7 = 0 using both factoring and the quadratic formula. You've seen that factoring is quicker when it works, but the quadratic formula is the reliable friend that always has your back. Practice both methods, and you'll be a quadratic equation master in no time! Mastering both factoring and the quadratic formula provides you with a versatile toolkit for solving quadratic equations. Factoring is often the quicker method when the equation is easily factorable, but the quadratic formula is the reliable fallback that works for all quadratic equations, regardless of their complexity. By understanding and practicing both methods, you'll develop a deeper appreciation for the structure and properties of quadratic equations. You'll also gain the ability to choose the most efficient method for a given problem, saving time and effort. Furthermore, mastering these techniques will build a strong foundation for more advanced mathematical concepts, such as complex numbers, polynomial equations, and calculus. The skills you've gained in solving quadratic equations will serve you well in various fields, from physics and engineering to economics and computer science. So, keep practicing, keep exploring, and keep challenging yourself with new problems. The more you work with quadratic equations, the more confident and proficient you'll become. And remember, mathematics is not just about finding the right answers; it's about developing critical thinking skills, problem-solving abilities, and a deeper understanding of the world around us. So, embrace the challenge, enjoy the journey, and keep mastering those quadratic equations!
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