Solving Quadratic Equations By Factoring A Step By Step Guide
Introduction: Mastering Quadratic Equations Through Factoring
Factoring quadratic equations is a fundamental skill in algebra, essential for solving various mathematical problems. This guide will provide a comprehensive walkthrough on how to solve the quadratic equation r² + 11r + 30 = 0 by factoring. Understanding the process of factoring allows us to break down complex expressions into simpler components, making it easier to find the solutions, also known as the roots or zeros, of the equation. This method is particularly useful when dealing with equations that can be neatly factored into binomial expressions. Before diving into the specifics of this equation, let's briefly touch on the general form of a quadratic equation and why factoring is such a valuable technique.
A quadratic equation is typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The solutions to this equation are the values of x that make the equation true. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring is often the quickest and most straightforward method when the quadratic expression can be easily factored. This approach not only helps in finding solutions but also enhances understanding of the relationship between the roots and the coefficients of the quadratic equation. Factoring a quadratic equation involves rewriting it as a product of two binomials. For instance, the quadratic expression x² + 5x + 6 can be factored into (x + 2)(x + 3). Setting each factor equal to zero allows us to find the values of x that satisfy the equation. This principle is the cornerstone of solving quadratic equations by factoring. In this guide, we will apply this principle to the equation r² + 11r + 30 = 0, providing a step-by-step explanation to ensure clarity and understanding.
We will start by identifying the coefficients in the given equation and then proceed to find two numbers that meet the factoring criteria. These numbers will be crucial in rewriting the quadratic expression into its factored form. Once we have the factored form, we will set each factor equal to zero and solve for r. These solutions will be the roots of the equation. By the end of this guide, you will have a firm grasp of how to factor quadratic equations and solve them efficiently. This skill is not only valuable for academic purposes but also for real-world applications where quadratic equations arise, such as in physics, engineering, and economics. Let's embark on this journey of mastering quadratic equations through factoring, starting with the equation r² + 11r + 30 = 0.
Step 1: Understanding the Quadratic Equation r² + 11r + 30 = 0
The first step in solving the quadratic equation r² + 11r + 30 = 0 by factoring involves a clear understanding of the equation's structure. Quadratic equations are characterized by their general form, ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. In our case, the variable is r, and the equation is r² + 11r + 30 = 0. Identifying the coefficients a, b, and c is crucial because these values dictate how we approach the factoring process. This initial step sets the stage for the subsequent steps, ensuring that we apply the correct methods and strategies to find the solutions.
In the given equation, r² + 11r + 30 = 0, we can identify the coefficients as follows: a is the coefficient of the r² term, which is 1 (since r² is the same as 1r²); b is the coefficient of the r term, which is 11; and c is the constant term, which is 30. Understanding these coefficients is the foundation for factoring the quadratic expression. The coefficient a tells us about the leading term, b is associated with the linear term, and c is the constant term that influences the final factored form. The goal of factoring is to rewrite the quadratic expression as a product of two binomials. This involves finding two numbers that, when multiplied, give us the constant term (c) and, when added, give us the coefficient of the linear term (b). In our case, we need two numbers that multiply to 30 and add up to 11. These numbers will play a key role in decomposing the middle term and expressing the quadratic expression in a factorable form. Identifying the coefficients correctly and understanding their significance is a critical step in solving quadratic equations by factoring. It lays the groundwork for the rest of the solution process, ensuring that we choose the right numbers and apply the correct factoring techniques. This foundational understanding makes the subsequent steps more intuitive and manageable, leading to an accurate solution.
Step 2: Finding the Factors
Now that we understand the structure of the quadratic equation, r² + 11r + 30 = 0, and have identified the coefficients a = 1, b = 11, and c = 30, the next crucial step is to find the right factors. The core of factoring a quadratic equation like this lies in identifying two numbers that meet specific criteria: their product must equal the constant term (c), and their sum must equal the coefficient of the linear term (b). This step is often the most challenging part of the factoring process, but with a systematic approach, it can be simplified. Finding these numbers allows us to rewrite the quadratic expression in a factorable form, ultimately leading to the solutions of the equation.
In our case, we are looking for two numbers that multiply to 30 and add up to 11. A methodical way to approach this is to list the factor pairs of 30 and then check which pair adds up to 11. The factor pairs of 30 are: (1, 30), (2, 15), (3, 10), and (5, 6). By examining these pairs, we can quickly determine that the pair (5, 6) satisfies both conditions: 5 multiplied by 6 equals 30, and 5 plus 6 equals 11. These numbers are the key to factoring the quadratic expression. Once we have identified these numbers, we can use them to rewrite the middle term (11r) as the sum of two terms (5r and 6r), which will allow us to factor by grouping. Finding the correct factors is not just a mathematical step; it's also a strategic one. It requires a bit of mental math and a keen eye for number relationships. However, with practice, identifying these factors becomes more intuitive. This step is the bridge between understanding the equation's structure and expressing it in a form that we can easily solve. Once we have the factors, we can move on to the next step, which involves rewriting the quadratic expression and factoring it into two binomials.
Step 3: Factoring the Quadratic Expression
With the factors identified as 5 and 6, the next step is to factor the quadratic expression r² + 11r + 30. Factoring involves rewriting the quadratic expression as a product of two binomials. This is achieved by using the factors we found in the previous step to decompose the middle term. This decomposition allows us to group terms and factor out common factors, ultimately leading to the factored form of the expression. The ability to factor quadratic expressions is a fundamental skill in algebra, providing a direct path to solving quadratic equations.
To begin, we rewrite the middle term, 11r, using the factors 5 and 6. The equation r² + 11r + 30 becomes r² + 5r + 6r + 30. This step is crucial because it sets up the expression for factoring by grouping. Now, we group the first two terms and the last two terms: (r² + 5r) + (6r + 30). Next, we factor out the greatest common factor (GCF) from each group. From the first group (r² + 5r), the GCF is r, so we factor out r to get r(r + 5). From the second group (6r + 30), the GCF is 6, so we factor out 6 to get 6(r + 5). Now, our expression looks like this: r(r + 5) + 6(r + 5). Notice that both terms have a common binomial factor, which is (r + 5). We factor out this common binomial factor to get (r + 5)(r + 6). This is the factored form of the quadratic expression r² + 11r + 30. Factoring the quadratic expression transforms it from a sum of terms into a product of binomials. This transformation is key because it allows us to use the zero-product property to find the solutions of the equation. The factored form (r + 5)(r + 6) provides a clear pathway to determining the values of r that make the equation equal to zero. This step is the heart of solving quadratic equations by factoring, and mastering it is essential for success in algebra.
Step 4: Applying the Zero-Product Property
After successfully factoring the quadratic expression r² + 11r + 30 into (r + 5)(r + 6), the next critical step is to apply the zero-product property. This property is a cornerstone of solving equations by factoring. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to take the factored form of the equation and break it down into simpler equations, each of which can be solved independently. Applying the zero-product property is what ultimately leads us to the solutions of the quadratic equation.
In our case, we have the equation (r + 5)(r + 6) = 0. According to the zero-product property, this equation is true if either (r + 5) = 0 or (r + 6) = 0, or both. This understanding transforms our single quadratic equation into two linear equations that are much easier to solve. We can now set each factor equal to zero and solve for r. For the first factor, we have r + 5 = 0. To solve for r, we subtract 5 from both sides of the equation, which gives us r = -5. For the second factor, we have r + 6 = 0. Similarly, we subtract 6 from both sides to solve for r, which gives us r = -6. Thus, we have found two solutions for r: r = -5 and r = -6. These are the values of r that make the original quadratic equation r² + 11r + 30 = 0 true. Applying the zero-product property is a pivotal step in the factoring process. It is the bridge between the factored form of the equation and the solutions. Without this property, we would not be able to derive the individual solutions from the product of factors. This step highlights the power of factoring as a method for solving quadratic equations, transforming a complex problem into a series of simpler ones.
Step 5: Finding the Solutions for r
Having applied the zero-product property to the factored equation (r + 5)(r + 6) = 0, we have arrived at the final step: finding the solutions for r. This step involves solving the two linear equations that resulted from applying the zero-product property. These solutions are the values of r that make the original quadratic equation true. Determining these solutions is the ultimate goal of solving any equation, and in the case of quadratic equations, it provides us with the points where the parabola intersects the x-axis.
From the zero-product property, we derived two equations: r + 5 = 0 and r + 6 = 0. To solve the first equation, r + 5 = 0, we subtract 5 from both sides. This isolates r on one side of the equation and gives us r = -5. This is one of the solutions to the quadratic equation. To solve the second equation, r + 6 = 0, we subtract 6 from both sides. This isolates r and gives us r = -6. This is the second solution to the quadratic equation. Therefore, the solutions for the equation r² + 11r + 30 = 0 are r = -5 and r = -6. These solutions can be written as a set {-5, -6}. It's important to verify these solutions by substituting them back into the original equation to ensure they make the equation true. Substituting r = -5 into r² + 11r + 30 = 0, we get (-5)² + 11(-5) + 30 = 25 - 55 + 30 = 0, which is true. Similarly, substituting r = -6 into the equation, we get (-6)² + 11(-6) + 30 = 36 - 66 + 30 = 0, which is also true. Thus, our solutions are correct. Finding the solutions for r is the culmination of the factoring process. It provides the specific values that satisfy the quadratic equation, giving us a complete understanding of its behavior. These solutions are not just numbers; they represent the points where the graph of the quadratic function intersects the x-axis, providing valuable insights into the function's properties.
Conclusion: Summarizing the Solution and the Importance of Factoring
In this comprehensive guide, we have successfully solved the quadratic equation r² + 11r + 30 = 0 by factoring. We began by understanding the structure of the equation and identifying the coefficients. Then, we found the factors that would allow us to rewrite the quadratic expression as a product of two binomials. This involved identifying two numbers that multiply to 30 and add up to 11, which we determined to be 5 and 6. Next, we factored the quadratic expression, transforming r² + 11r + 30 into (r + 5)(r + 6). We then applied the zero-product property, which allowed us to set each factor equal to zero and solve for r. This led us to the solutions r = -5 and r = -6. These solutions are the values of r that make the original equation true, and they represent the roots of the quadratic equation. By following these steps, we have demonstrated the power and efficiency of factoring as a method for solving quadratic equations.
Factoring is a fundamental skill in algebra, and its importance extends far beyond solving simple quadratic equations. It is a crucial technique for simplifying algebraic expressions, solving higher-degree polynomial equations, and understanding the behavior of functions. Factoring also plays a significant role in calculus and other advanced mathematical topics. Mastering factoring enhances problem-solving skills and provides a deeper understanding of mathematical concepts. The ability to factor quadratic equations efficiently is a valuable asset in various fields, including engineering, physics, economics, and computer science. In conclusion, factoring is not just a method for solving equations; it is a gateway to more advanced mathematical concepts and applications. The process we followed in this guide—understanding the equation, finding the factors, factoring the expression, applying the zero-product property, and solving for the variable—is a systematic approach that can be applied to a wide range of quadratic equations. By mastering these steps, you will be well-equipped to tackle more complex mathematical problems and gain a greater appreciation for the beauty and power of algebra.
