Solving X² - 3x - 10 = 0 Find The Roots By Factorization
In mathematics, quadratic equations play a crucial role, appearing in various fields such as physics, engineering, and computer science. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation, also known as the solutions or zeros of the quadratic function. There are several methods to find these roots, including factorization, completing the square, and using the quadratic formula. This article focuses on finding the roots of the quadratic equation x² - 3x - 10 = 0 by factorization, providing a step-by-step guide to help you understand and apply this method effectively. Understanding factorization is fundamental in solving quadratic equations and provides a solid foundation for more advanced mathematical concepts. By mastering this technique, you will be well-equipped to tackle a wide range of problems in algebra and beyond. We will delve into the detailed process, explaining each step clearly and offering additional insights to ensure a comprehensive understanding. This approach not only helps in solving this particular equation but also equips you with a versatile skill applicable to various quadratic equations. So, let's embark on this mathematical journey to unravel the roots of x² - 3x - 10 = 0 through factorization.
Understanding Quadratic Equations
Before diving into the specifics of factorization, it's essential to understand what a quadratic equation is and its general form. A quadratic equation is defined as an equation of the form ax² + bx + c = 0, where a, b, and c are constants, with a not equal to zero. The coefficients a, b, and c determine the shape and position of the parabola when the quadratic equation is graphed, and they also influence the roots of the equation. The term ax² is the quadratic term, bx is the linear term, and c is the constant term. The roots of the quadratic equation are the values of x that make the equation true, i.e., the values of x that satisfy ax² + bx + c = 0. These roots represent the points where the parabola intersects the x-axis on a graph. A quadratic equation can have two real roots, one real root (which is a repeated root), or two complex roots, depending on the discriminant (b² - 4ac). The discriminant provides valuable information about the nature of the roots: if b² - 4ac > 0, there are two distinct real roots; if b² - 4ac = 0, there is exactly one real root; and if b² - 4ac < 0, there are two complex roots. Understanding these foundational concepts is crucial for effectively solving quadratic equations. In the equation x² - 3x - 10 = 0, a = 1, b = -3, and c = -10. Recognizing these coefficients is the first step in applying various methods to find the roots, including factorization. Grasping the fundamentals of quadratic equations ensures a smoother transition into more advanced techniques and problem-solving scenarios. Thus, before we proceed with factorization, ensuring a solid understanding of the general form and its components is paramount.
Factorization Method: A Step-by-Step Guide
The factorization method is a technique used to solve quadratic equations by breaking down the quadratic expression into a product of two linear expressions. This method is particularly effective when the quadratic equation can be easily factored. The main idea behind factorization is to rewrite the quadratic expression ax² + bx + c as (px + q)(rx + s), where p, q, r, and s are constants. Once the quadratic expression is factored, we can find the roots by setting each factor equal to zero and solving for x. This step-by-step guide will walk you through the process of factoring the given quadratic equation x² - 3x - 10 = 0. First, identify the coefficients a, b, and c. In this case, a = 1, b = -3, and c = -10. The next step is to find two numbers that multiply to give ac and add up to b. For our equation, we need two numbers that multiply to (1)(-10) = -10 and add up to -3. These numbers are -5 and 2 because (-5)(2) = -10 and -5 + 2 = -3. Now, rewrite the middle term -3x using these two numbers: x² - 5x + 2x - 10 = 0. Next, group the terms in pairs: (x² - 5x) + (2x - 10) = 0. Factor out the greatest common factor (GCF) from each pair: x(x - 5) + 2(x - 5) = 0. Notice that (x - 5) is a common factor in both terms. Factor out the common factor (x - 5): (x - 5)(x + 2) = 0. Finally, set each factor equal to zero and solve for x: x - 5 = 0 or x + 2 = 0. Solving these linear equations gives the roots of the quadratic equation. By following these steps, you can effectively factor quadratic equations and find their roots. Understanding each step and its underlying logic is crucial for mastering this method.
Step 1: Identify Coefficients
The first crucial step in solving a quadratic equation by factorization is to identify the coefficients a, b, and c in the standard form ax² + bx + c = 0. This identification is the foundation for applying the factorization technique effectively. In our given equation, x² - 3x - 10 = 0, we can easily determine the coefficients by comparing it with the standard 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 -3. The constant term c is the term without any x, which is -10. Therefore, we have a = 1, b = -3, and c = -10. Correctly identifying these coefficients is essential because they are used in the subsequent steps of the factorization process. Misidentifying them can lead to incorrect factoring and ultimately, wrong roots. For instance, if we incorrectly identified b as 3 instead of -3, the whole factorization process would be flawed. Moreover, understanding the role of each coefficient provides insights into the nature of the quadratic equation. The coefficient a determines the direction and width of the parabola, while b and c influence its position on the coordinate plane. Grasping the significance of each coefficient enhances your ability to analyze and solve quadratic equations more efficiently. In summary, the accurate identification of a, b, and c sets the stage for a successful factorization, making it a critical initial step in finding the roots of a quadratic equation. Mastering this step ensures a smoother and more accurate problem-solving process.
Step 2: Find Two Numbers
After identifying the coefficients a, b, and c, the next critical step in factorization is to find two numbers that satisfy specific conditions related to these coefficients. These two numbers are essential for rewriting the middle term of the quadratic equation, which is a key part of the factoring process. Specifically, we need to find two numbers such that their product equals ac and their sum equals b. In our equation, x² - 3x - 10 = 0, we have already identified a = 1, b = -3, and c = -10. Therefore, we need to find two numbers that multiply to ac = (1)(-10) = -10 and add up to b = -3. This step often involves some trial and error, but a systematic approach can make it more manageable. Start by listing pairs of factors of ac and then check which pair adds up to b. For -10, the pairs of factors are: (1, -10), (-1, 10), (2, -5), and (-2, 5). Now, we check the sum of each pair: 1 + (-10) = -9, -1 + 10 = 9, 2 + (-5) = -3, and -2 + 5 = 3. We can see that the pair (2, -5) satisfies both conditions: their product is -10, and their sum is -3. These two numbers, 2 and -5, are crucial for the next step in the factorization process. Finding the correct pair of numbers is a pivotal moment in factorization, as it determines the success of the subsequent steps. A clear understanding of this step allows you to efficiently break down the quadratic expression into manageable parts. By mastering this technique, you will be able to quickly identify the correct numbers and proceed with the factorization.
Step 3: Rewrite the Middle Term
Once we have identified the two numbers that multiply to ac and add up to b, the next step is to rewrite the middle term of the quadratic equation using these numbers. This process is fundamental in transforming the quadratic expression into a form that can be easily factored. In our equation, x² - 3x - 10 = 0, we found that the two numbers are -5 and 2. The middle term is -3x, and we will rewrite it as the sum of two terms using these numbers. Specifically, we replace -3x with -5x + 2x. So, the equation x² - 3x - 10 = 0 becomes x² - 5x + 2x - 10 = 0. Rewriting the middle term in this way allows us to group terms and factor by grouping, which is a standard technique in factorization. The key to this step is ensuring that the new expression is equivalent to the original one. In other words, -5x + 2x must equal -3x. If there is any discrepancy, the subsequent steps will lead to an incorrect solution. This rewriting process effectively breaks down the quadratic expression into four terms, setting the stage for the next phase of factorization. It’s crucial to double-check that the rewritten expression maintains the original equation's integrity. The act of rewriting the middle term is a strategic maneuver that simplifies the factorization process, making it easier to identify common factors and arrive at the solutions. By mastering this step, you lay a solid foundation for successfully factoring quadratic equations.
Step 4: Factor by Grouping
After rewriting the middle term, the next step in the factorization process is to factor by grouping. This technique involves grouping the terms in pairs and factoring out the greatest common factor (GCF) from each pair. In our equation, which has been rewritten as x² - 5x + 2x - 10 = 0, we group the terms as (x² - 5x) + (2x - 10) = 0. Now, we factor out the GCF from each pair. In the first group, (x² - 5x), the GCF is x. Factoring out x gives us x(x - 5). In the second group, (2x - 10), the GCF is 2. Factoring out 2 gives us 2(x - 5). So, the equation becomes x(x - 5) + 2(x - 5) = 0. The purpose of factoring by grouping is to identify a common binomial factor. In this case, we can see that (x - 5) is a common factor in both terms. This common binomial factor is the key to the next step. Factoring by grouping is a powerful technique because it transforms a four-term expression into a product of two binomials. This transformation is essential for solving the quadratic equation. If the grouping is done correctly, a common binomial factor should always emerge. If a common factor is not apparent, it may indicate an error in the previous steps, such as incorrect identification of the two numbers or improper rewriting of the middle term. Mastering the technique of factoring by grouping is crucial for solving a wide range of quadratic equations. It simplifies the process and brings us closer to finding the roots of the equation. Thus, understanding and practicing this step is vital for successfully factoring quadratic expressions.
Step 5: Factor Out the Common Factor
Following the factoring by grouping step, the next crucial action is to factor out the common factor. This step consolidates the expression into a product of two factors, making it easier to find the roots of the quadratic equation. After factoring by grouping in our equation, x² - 3x - 10 = 0, we arrived at the expression x(x - 5) + 2(x - 5) = 0. Here, we observe that (x - 5) is a common factor in both terms. Factoring out the common factor (x - 5), we get (x - 5)(x + 2) = 0. This step is critical because it transforms the equation into a form where we have a product of two factors equal to zero. This sets the stage for using 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. The process of factoring out the common factor effectively simplifies the equation, making it straightforward to identify the roots. It is essential to ensure that the common factor is factored out correctly, as any error in this step will lead to incorrect solutions. This step also highlights the elegance of the factorization method, where complex expressions are broken down into simpler, manageable components. By mastering this technique, you can efficiently handle a variety of quadratic equations. The ability to recognize and factor out common factors is a fundamental skill in algebra, applicable not only to quadratic equations but also to other areas of mathematics. Therefore, a thorough understanding of this step is vital for success in problem-solving.
Step 6: Apply the Zero-Product Property
Once the quadratic equation is factored into the form (x - 5)(x + 2) = 0, the next vital step is to apply the zero-product property. This property is a cornerstone in solving equations by factoring and states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if AB = 0, then either A = 0 or B = 0 (or both). Applying this property to our factored equation, (x - 5)(x + 2) = 0, we set each factor equal to zero: x - 5 = 0 or x + 2 = 0. This step transforms the single quadratic equation into two simpler linear equations, which are much easier to solve. The zero-product property is a powerful tool because it breaks down a complex problem into smaller, manageable parts. It is a direct consequence of the properties of multiplication and zero, making it a fundamental concept in algebra. Correctly applying this property is essential for finding the roots of the quadratic equation. Any mistake in setting the factors equal to zero will lead to incorrect solutions. This step is also a clear demonstration of why factorization is such an effective method for solving quadratic equations. By factoring the equation, we create a situation where the zero-product property can be applied, simplifying the process of finding the roots. Understanding and applying the zero-product property is crucial not only for solving quadratic equations but also for various other algebraic problems. It is a versatile tool that simplifies many types of equations, making it a core concept in mathematics.
Step 7: Solve for x
After applying the zero-product property, the final step in the factorization method is to solve for x. This involves solving the linear equations that result from setting each factor equal to zero. In our example, after applying the zero-product property to (x - 5)(x + 2) = 0, we obtained two linear equations: x - 5 = 0 and x + 2 = 0. To solve the first equation, x - 5 = 0, we add 5 to both sides, which gives us x = 5. To solve the second equation, x + 2 = 0, we subtract 2 from both sides, which gives us x = -2. Therefore, the roots of the quadratic equation x² - 3x - 10 = 0 are x = 5 and x = -2. These values of x are the solutions to the original quadratic equation, meaning that when we substitute these values back into the equation, it will hold true. Verifying the solutions by substituting them back into the original equation is a good practice to ensure accuracy. Solving for x involves straightforward algebraic manipulation, but it is a crucial step as it provides the final answer. The solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph. Understanding how to solve for x after factoring is essential for completing the factorization method. This step is a culmination of all the previous steps, and it provides the concrete values that satisfy the original equation. Mastering this step ensures that you can successfully find the roots of quadratic equations using factorization.
Roots of the Equation x² - 3x - 10 = 0
By following the steps of the factorization method, we have successfully found the roots of the equation x² - 3x - 10 = 0. The roots are the values of x that make the equation true, and in this case, we found two distinct real roots. Through the process of factorization, we rewrote the equation as (x - 5)(x + 2) = 0. Applying the zero-product property, we set each factor equal to zero, resulting in two linear equations: x - 5 = 0 and x + 2 = 0. Solving these equations, we found x = 5 and x = -2. Therefore, the roots of the quadratic equation x² - 3x - 10 = 0 are 5 and -2. These roots represent the points where the graph of the quadratic function y = x² - 3x - 10 intersects the x-axis. To verify these solutions, we can substitute them back into the original equation. For x = 5, we have: (5)² - 3(5) - 10 = 25 - 15 - 10 = 0, which confirms that 5 is a root. For x = -2, we have: (-2)² - 3(-2) - 10 = 4 + 6 - 10 = 0, which confirms that -2 is also a root. Finding the roots of a quadratic equation is a fundamental concept in algebra, and the factorization method is one of the key techniques for achieving this. The roots provide valuable information about the behavior of the quadratic function and its graphical representation. In summary, the roots of the equation x² - 3x - 10 = 0 are 5 and -2, and we have verified these solutions by substitution. This comprehensive process demonstrates the effectiveness of the factorization method in solving quadratic equations.
In conclusion, we have successfully found the roots of the quadratic equation x² - 3x - 10 = 0 using the factorization method. This method involves breaking down the quadratic expression into a product of two linear expressions and then applying the zero-product property to find the solutions. We began by identifying the coefficients a, b, and c in the equation, which were 1, -3, and -10, respectively. Then, we found two numbers that multiply to ac = -10 and add up to b = -3, which were -5 and 2. We rewrote the middle term using these numbers, transforming the equation into x² - 5x + 2x - 10 = 0. Next, we factored by grouping, which led to the expression x(x - 5) + 2(x - 5) = 0. We then factored out the common factor (x - 5), resulting in (x - 5)(x + 2) = 0. Applying the zero-product property, we set each factor equal to zero: x - 5 = 0 and x + 2 = 0. Solving these linear equations, we found the roots x = 5 and x = -2. These roots were verified by substituting them back into the original equation, confirming their validity. The factorization method is a powerful tool for solving quadratic equations, especially when the equation can be easily factored. It provides a clear and systematic approach to finding the roots, which are crucial for understanding the behavior and graphical representation of quadratic functions. By mastering this method, you can effectively solve a wide range of quadratic equations and apply this knowledge to more advanced mathematical concepts. Understanding each step of the factorization process enhances your problem-solving skills and strengthens your foundation in algebra. This comprehensive guide demonstrates the step-by-step process of using factorization to solve quadratic equations, making it a valuable resource for students and anyone interested in mathematics.
