Solving Quadratic Equations Step-by-Step Guide With Examples

This comprehensive guide delves into the fascinating world of quadratic equations, providing step-by-step solutions and explanations for a variety of examples. Quadratic equations, characterized by their highest power of 2, are fundamental in mathematics and have wide-ranging applications in various fields, including physics, engineering, and economics. This article will methodically dissect five distinct quadratic equations, illustrating different approaches to finding their roots (or solutions). We will explore techniques such as factoring, which involves breaking down the quadratic expression into simpler factors, and the quadratic formula, a universal tool for solving any quadratic equation. By understanding these methods, you'll gain a solid foundation for tackling more complex mathematical problems. So, let's embark on this journey of mathematical exploration and unlock the secrets of quadratic equations. Understanding quadratic equations is crucial for success in higher-level mathematics and related disciplines. Mastering these concepts will empower you to solve a wide array of problems and make informed decisions in various real-world scenarios.

1) x² + 11x + 24 = 0

Let's begin with the first equation: x² + 11x + 24 = 0. Our goal is to find the values of 'x' that satisfy this equation. The most common method for solving such equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. To factor x² + 11x + 24, we need to find two numbers that add up to 11 (the coefficient of the 'x' term) and multiply to 24 (the constant term). These numbers are 3 and 8, since 3 + 8 = 11 and 3 * 8 = 24. Therefore, we can rewrite the equation as (x + 3)(x + 8) = 0. Now, for the product of two factors to be zero, at least one of them must be zero. This gives us two possibilities: x + 3 = 0 or x + 8 = 0. Solving these linear equations, we get x = -3 and x = -8. These are the roots of the quadratic equation x² + 11x + 24 = 0. We can verify these solutions by substituting them back into the original equation. For x = -3, we have (-3)² + 11(-3) + 24 = 9 - 33 + 24 = 0. For x = -8, we have (-8)² + 11(-8) + 24 = 64 - 88 + 24 = 0. Both solutions satisfy the equation, confirming that they are the correct roots. Factoring quadratic equations is a fundamental skill in algebra, and this example demonstrates the process clearly. Understanding how to find the correct factors is essential for solving a wide range of quadratic equations efficiently. This particular equation lends itself well to factoring because the coefficients are relatively small and the factors are integers. However, not all quadratic equations can be easily factored, which is where other methods, such as the quadratic formula, come into play. The ability to recognize factorable quadratics and apply the factoring technique is a valuable asset in mathematical problem-solving.

2) x² - 2x - 15 = 0

Next, let's tackle the equation x² - 2x - 15 = 0. Again, we can attempt to solve this by factoring. We need to find two numbers that add up to -2 (the coefficient of the 'x' term) and multiply to -15 (the constant term). These numbers are -5 and 3, since -5 + 3 = -2 and -5 * 3 = -15. So, we can factor the equation as (x - 5)(x + 3) = 0. Setting each factor equal to zero gives us x - 5 = 0 or x + 3 = 0. Solving these equations, we find x = 5 and x = -3. These are the roots of the quadratic equation x² - 2x - 15 = 0. To verify, we substitute these values back into the original equation. For x = 5, we have (5)² - 2(5) - 15 = 25 - 10 - 15 = 0. For x = -3, we have (-3)² - 2(-3) - 15 = 9 + 6 - 15 = 0. Both solutions satisfy the equation, confirming their validity. This example further illustrates the power of factoring in solving quadratic equations. The process of identifying the correct factors is crucial, and it often involves some trial and error. Practicing factoring with various examples will help you develop the intuition and skills needed to solve quadratic equations efficiently. It's also important to note that the signs of the factors are critical. In this case, one factor was negative and the other was positive, which is necessary when the constant term is negative. This distinction helps narrow down the possible factor combinations. While factoring is a powerful technique, it's not always applicable. When factoring becomes difficult or impossible, the quadratic formula provides a reliable alternative.

3) x² - 17x + 15 = 0

Now, let's consider the equation x² - 17x + 15 = 0. In this case, factoring might not be as straightforward as in the previous examples. It's challenging to find two integers that add up to -17 and multiply to 15. Therefore, we'll employ the quadratic formula, a general solution for any quadratic equation of the form ax² + bx + c = 0. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. In our equation, a = 1, b = -17, and c = 15. Substituting these values into the quadratic formula, we get: x = (17 ± √((-17)² - 4 * 1 * 15)) / (2 * 1). Simplifying the expression under the square root, we have: (-17)² - 4 * 1 * 15 = 289 - 60 = 229. So, the equation becomes: x = (17 ± √229) / 2. This gives us two solutions: x = (17 + √229) / 2 and x = (17 - √229) / 2. These solutions are irrational numbers, which means they cannot be expressed as a simple fraction. We can approximate these values using a calculator: x ≈ (17 + 15.13) / 2 ≈ 16.065 and x ≈ (17 - 15.13) / 2 ≈ 0.935. These are the approximate roots of the equation x² - 17x + 15 = 0. This example demonstrates the power and versatility of the quadratic formula. The quadratic formula is a fundamental tool for solving quadratic equations, especially when factoring is not feasible. Understanding its derivation and application is crucial for any student of algebra. The discriminant (b² - 4ac) within the formula provides valuable information about the nature of the roots. In this case, the discriminant is positive (229), indicating that the equation has two distinct real roots. If the discriminant were zero, the equation would have one real root (a repeated root), and if it were negative, the equation would have two complex roots. The quadratic formula ensures that we can find the solutions to any quadratic equation, regardless of the nature of its roots.

4) x² + x - 72 = 0

Let's move on to the equation x² + x - 72 = 0. We can attempt to solve this by factoring. We need to find two numbers that add up to 1 (the coefficient of the 'x' term) and multiply to -72 (the constant term). These numbers are 9 and -8, since 9 + (-8) = 1 and 9 * (-8) = -72. Therefore, we can factor the equation as (x + 9)(x - 8) = 0. Setting each factor equal to zero gives us x + 9 = 0 or x - 8 = 0. Solving these equations, we find x = -9 and x = 8. These are the roots of the quadratic equation x² + x - 72 = 0. To verify, we substitute these values back into the original equation. For x = -9, we have (-9)² + (-9) - 72 = 81 - 9 - 72 = 0. For x = 8, we have (8)² + 8 - 72 = 64 + 8 - 72 = 0. Both solutions satisfy the equation, confirming their validity. This example reinforces the factoring technique and demonstrates how to handle negative constant terms. Factoring is often the quickest way to solve quadratic equations when it's applicable. Recognizing the patterns and relationships between the coefficients can significantly speed up the factoring process. In this case, the relatively large constant term (-72) might seem daunting at first, but breaking it down into its factors (9 and -8) reveals the solution. This highlights the importance of number sense and the ability to mentally manipulate factors. As with previous examples, it's always a good practice to verify the solutions by substituting them back into the original equation. This helps catch any errors and ensures that the roots are correct. While factoring is efficient for this equation, the quadratic formula could also be used as an alternative method.

5) x² - x - 56 = 0

Finally, let's consider the equation x² - x - 56 = 0. We can try to solve this equation by factoring. We need to find two numbers that add up to -1 (the coefficient of the 'x' term) and multiply to -56 (the constant term). These numbers are -8 and 7, since -8 + 7 = -1 and -8 * 7 = -56. Therefore, we can factor the equation as (x - 8)(x + 7) = 0. Setting each factor equal to zero gives us x - 8 = 0 or x + 7 = 0. Solving these equations, we find x = 8 and x = -7. These are the roots of the quadratic equation x² - x - 56 = 0. To verify, we substitute these values back into the original equation. For x = 8, we have (8)² - (8) - 56 = 64 - 8 - 56 = 0. For x = -7, we have (-7)² - (-7) - 56 = 49 + 7 - 56 = 0. Both solutions satisfy the equation, confirming their validity. This final example further solidifies the factoring technique and demonstrates its effectiveness in solving quadratic equations. Mastering factoring requires practice and an understanding of number relationships. The ability to quickly identify factors that satisfy the required conditions is a valuable skill in algebra. This example, like the others, underscores the importance of checking solutions to ensure accuracy. The combination of factoring and verification provides a robust approach to solving quadratic equations. While the quadratic formula can always be used as a fallback method, factoring often provides a more efficient solution when applicable. By working through these examples, you've gained a solid understanding of how to solve quadratic equations using factoring and the quadratic formula, equipping you with the tools to tackle a wide range of mathematical problems.

In conclusion, this exploration of quadratic equations has demonstrated the power and versatility of different solution techniques. From factoring to the quadratic formula, we've seen how to systematically find the roots of these fundamental mathematical expressions. Understanding quadratic equations is not just about memorizing formulas; it's about developing a deep conceptual understanding and the ability to apply the appropriate methods to solve problems efficiently. The examples discussed in this article provide a solid foundation for further exploration of more advanced mathematical topics. Remember to practice regularly and always verify your solutions to ensure accuracy. With dedication and a clear understanding of the concepts, you can master the art of solving quadratic equations and unlock their many applications in various fields.