Solving Quadratic Equations 15x² = 540, (3x - 3)² = 36, And X² = 121

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Introduction

In this comprehensive guide, we will delve into the methods for solving quadratic equations, focusing on three specific examples: 1) 15x² = 540, 2) (3x - 3)² = 36, and 3) x² = 121. Quadratic equations are polynomial equations of the second degree, meaning they contain at least one term that is squared. These types of equations appear frequently in various fields, including mathematics, physics, engineering, and economics. Understanding how to solve them is a fundamental skill in algebra. This guide aims to provide a clear, step-by-step approach to solving these equations, ensuring that readers can confidently tackle similar problems in the future. We will cover the necessary algebraic manipulations and techniques to find the solutions, also known as roots, of these equations. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will help you master the art of solving quadratic equations. Let’s begin by understanding the basic structure of a quadratic equation and the common methods used to solve them. This groundwork will allow us to efficiently address the specific examples presented, providing a solid understanding of the underlying principles and fostering confidence in your problem-solving abilities.

Understanding Quadratic Equations

A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. The coefficient 'a' cannot be zero; otherwise, the equation becomes linear. Solving a quadratic equation involves finding the values of x that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of equations. For instance, factoring is efficient when the equation can be easily factored, while the quadratic formula is a more general approach that works for all quadratic equations. Completing the square is another method that is useful for understanding the derivation of the quadratic formula and for solving equations that do not factor easily. Before diving into the specific examples, it’s important to have a firm grasp of these methods. Factoring involves breaking down the quadratic expression into a product of two binomials, which can then be solved by setting each factor equal to zero. The quadratic formula, given by x = [-b ± √(b² - 4ac)] / (2a), is a universal tool that provides solutions regardless of the complexity of the coefficients. Completing the square involves manipulating the equation to form a perfect square trinomial, which simplifies the process of finding the solutions. Understanding these methods will not only help you solve the given equations but also provide a strong foundation for tackling more complex algebraic problems. Let's delve deeper into each method before applying them to our specific examples.

Methods to Solve Quadratic Equations

Factoring

Factoring is a method used to simplify expressions and solve equations. In the context of quadratic equations, factoring involves breaking down the quadratic expression into a product of two binomials. For example, the equation x² + 5x + 6 = 0 can be factored into (x + 2)(x + 3) = 0. The solutions are then found by setting each factor equal to zero, which gives x = -2 and x = -3. This method is particularly effective when the quadratic expression can be easily factored, which often occurs when the coefficients are integers and the roots are rational numbers. However, not all quadratic equations can be factored easily, and in such cases, alternative methods like the quadratic formula or completing the square are more appropriate. The key to successful factoring lies in recognizing patterns and applying algebraic principles. Common factoring techniques include identifying the greatest common factor (GCF), using the difference of squares formula (a² - b² = (a + b)(a - b)), and recognizing perfect square trinomials (a² + 2ab + b² = (a + b)²). Practice and familiarity with these techniques will improve your ability to factor quadratic expressions quickly and accurately. Factoring is not only a valuable tool for solving quadratic equations but also a fundamental skill in simplifying algebraic expressions and solving various mathematical problems. It provides a deep understanding of the structure of polynomials and their relationships, making it an essential technique for any student of algebra.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by x = [-b ± √(b² - 4ac)] / (2a). This formula can be used to find the solutions of any quadratic equation, regardless of whether it can be factored easily or not. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If it is negative, there are two complex solutions. The quadratic formula is derived by completing the square on the general form of the quadratic equation, and it is a powerful tool that ensures a solution can always be found. To use the quadratic formula, simply identify the coefficients a, b, and c, substitute them into the formula, and simplify. The ± symbol indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it. The quadratic formula is not only useful for solving quadratic equations but also for analyzing the behavior of quadratic functions and their graphs. Understanding the discriminant helps in determining the number and type of roots, which can be crucial in various applications. Mastering the quadratic formula is an essential skill for anyone studying algebra and related fields, as it provides a reliable and efficient method for solving a wide range of problems.

Completing the Square

Completing the square is another method for solving quadratic equations, which involves transforming the equation into a perfect square trinomial. This method is particularly useful for understanding the derivation of the quadratic formula and for solving equations that do not factor easily. The process of completing the square involves several steps. First, ensure that the coefficient of x² is 1. If it is not, divide the entire equation by that coefficient. Next, move the constant term to the right side of the equation. Then, take half of the coefficient of the x term, square it, and add it to both sides of the equation. This step creates a perfect square trinomial on the left side of the equation, which can be factored into the form (x + k)² or (x - k)², where k is a constant. Finally, take the square root of both sides of the equation and solve for x. Completing the square not only provides a method for solving quadratic equations but also offers insights into the structure and properties of quadratic expressions. It demonstrates how any quadratic equation can be transformed into a form that is easily solvable, and it forms the basis for the derivation of the quadratic formula. This method is also valuable in various other contexts, such as finding the vertex of a parabola and solving optimization problems. By mastering completing the square, you gain a deeper understanding of quadratic equations and their applications, enhancing your problem-solving skills and algebraic intuition. This method, while sometimes more involved than factoring or using the quadratic formula, offers a comprehensive approach to dealing with quadratic equations and solidifies your understanding of algebraic manipulations.

Solving 15x² = 540

To solve the quadratic equation 15x² = 540, we will use algebraic manipulation to isolate x². The goal is to get x² by itself on one side of the equation, which will allow us to take the square root and find the values of x that satisfy the equation. This method is straightforward and efficient when dealing with equations in this form, where there is only one term involving x and a constant term. First, we divide both sides of the equation by 15 to isolate x². This gives us x² = 540 / 15, which simplifies to x² = 36. Now that we have x² isolated, we can take the square root of both sides of the equation to solve for x. Remember that when taking the square root, we must consider both the positive and negative roots, as both will satisfy the equation. This is because squaring a positive or a negative number will result in a positive number. Therefore, we have x = ±√36. The square root of 36 is 6, so the solutions are x = 6 and x = -6. These are the two values of x that, when substituted back into the original equation, will make the equation true. Solving quadratic equations in this manner is a fundamental algebraic skill, and understanding the steps involved is crucial for tackling more complex problems. Let’s walk through the steps again to reinforce the method. This approach of isolating the squared term and then taking the square root is a common technique used in solving various types of equations, making it an essential tool in your mathematical toolkit.

Step-by-Step Solution

  1. Divide both sides by 15: Starting with 15x² = 540, divide both sides by 15 to isolate x². This gives us x² = 540 / 15, which simplifies to x² = 36.
  2. Take the square root of both sides: To solve for x, take the square root of both sides of the equation. Remember to consider both positive and negative roots: x = ±√36.
  3. Simplify: The square root of 36 is 6, so we have x = ±6.
  4. Solutions: The solutions are x = 6 and x = -6.

Solving (3x - 3)² = 36

To solve the quadratic equation (3x - 3)² = 36, we will employ a method that involves taking the square root of both sides initially and then solving the resulting linear equations. This approach is particularly effective when the quadratic equation is given in the form of a squared expression equal to a constant. By taking the square root of both sides, we simplify the equation and can proceed to solve for x using basic algebraic principles. This method avoids the need to expand the squared term, which can sometimes lead to more complex calculations. The first step is to take the square root of both sides of the equation. When doing this, it's crucial to remember to consider both the positive and negative roots, as both will satisfy the original equation due to the squaring operation. This gives us two equations: 3x - 3 = 6 and 3x - 3 = -6. Next, we solve each of these linear equations separately. For the first equation, 3x - 3 = 6, we add 3 to both sides to isolate the term with x, resulting in 3x = 9. Then, we divide both sides by 3 to solve for x, giving us x = 3. For the second equation, 3x - 3 = -6, we again add 3 to both sides, which gives us 3x = -3. Dividing both sides by 3, we find x = -1. Therefore, the solutions to the original quadratic equation are x = 3 and x = -1. This step-by-step process illustrates how to effectively solve equations of this type, emphasizing the importance of considering both positive and negative roots and applying basic algebraic manipulations to isolate the variable. Let’s walk through the steps again to reinforce the method.

Step-by-Step Solution

  1. Take the square root of both sides: Starting with (3x - 3)² = 36, take the square root of both sides. Remember to consider both positive and negative roots: 3x - 3 = ±√36.
  2. Simplify: The square root of 36 is 6, so we have 3x - 3 = ±6.
  3. Set up two equations: This gives us two separate equations to solve: 3x - 3 = 6 and 3x - 3 = -6.
  4. Solve the first equation (3x - 3 = 6):
    • Add 3 to both sides: 3x = 9
    • Divide by 3: x = 3
  5. Solve the second equation (3x - 3 = -6):
    • Add 3 to both sides: 3x = -3
    • Divide by 3: x = -1
  6. Solutions: The solutions are x = 3 and x = -1.

Solving x² = 121

To solve the quadratic equation x² = 121, we will use the method of taking the square root of both sides. This is a straightforward approach for equations in this simple form, where we have x² equal to a constant. The key is to remember that when taking the square root of both sides, we must consider both the positive and negative roots. This is because squaring either a positive or a negative number will result in a positive value. Starting with x² = 121, we take the square root of both sides. This gives us x = ±√121. The square root of 121 is 11, so we have x = ±11. This means that there are two solutions to the equation: x = 11 and x = -11. Both of these values, when squared, will equal 121, satisfying the original equation. This method is a fundamental technique in algebra and is essential for solving various types of equations, including more complex quadratic equations. Understanding how to correctly apply this method ensures that all possible solutions are found. Let’s walk through the steps again to reinforce the method. The simplicity of this method makes it an efficient and reliable way to solve equations of this form, highlighting the importance of understanding the properties of square roots and their application in algebra.

Step-by-Step Solution

  1. Take the square root of both sides: Starting with x² = 121, take the square root of both sides. Remember to consider both positive and negative roots: x = ±√121.
  2. Simplify: The square root of 121 is 11, so we have x = ±11.
  3. Solutions: The solutions are x = 11 and x = -11.

Conclusion

In this guide, we have thoroughly examined how to solve three different quadratic equations: 15x² = 540, (3x - 3)² = 36, and x² = 121. We explored various methods, including isolating the variable and taking the square root, and considering both positive and negative roots. Each equation presented a unique scenario, allowing us to apply different techniques and reinforce the fundamental principles of solving quadratic equations. By breaking down each problem into step-by-step solutions, we demonstrated a clear and logical approach to problem-solving, ensuring that the methods are easily understandable and replicable. Quadratic equations are a crucial topic in algebra, and mastering the techniques to solve them is essential for further studies in mathematics and related fields. The ability to solve these equations is not only important for academic success but also has practical applications in various real-world scenarios. Throughout this guide, we emphasized the importance of understanding the underlying concepts and the reasoning behind each step. This approach fosters a deeper understanding of the material and enhances problem-solving skills, allowing you to tackle a wide range of mathematical challenges confidently. Whether you are a student learning algebra for the first time or someone looking to refresh your knowledge, this guide provides a solid foundation for understanding and solving quadratic equations. By practicing these methods and applying them to different problems, you will strengthen your algebraic skills and develop a greater appreciation for the power and elegance of mathematical problem-solving. We encourage you to revisit these examples and techniques as needed, and to continue exploring the vast and fascinating world of mathematics.