Solve The Equation 2x^2 = 8x.
In the realm of mathematics, particularly within the study of algebra, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place due to their frequent appearance in real-world applications and their mathematical elegance. This article delves into the process of finding the truth set, or the solution set, of a specific quadratic equation: 2x² = 8x. We will explore the step-by-step method to solve this equation, discuss the underlying principles, and highlight the importance of understanding quadratic equations in a broader mathematical context.
Understanding Quadratic Equations
Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of the variable (usually x) that satisfy the equation. These solutions represent the points where the quadratic function intersects the x-axis on a graph. Finding these solutions is a crucial aspect of solving quadratic equations.
There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific form of the equation and the ease with which it can be manipulated. In the case of 2x² = 8x, factoring proves to be a straightforward and efficient approach. Understanding the different methods and their applicability is essential for mastering quadratic equations.
Solving 2x² = 8x by Factoring
The equation we aim to solve is 2x² = 8x. The first step in solving this quadratic equation by factoring is to rearrange the equation into the standard form, which is ax² + bx + c = 0. To do this, we subtract 8x from both sides of the equation:
2x² - 8x = 0
Now that the equation is in the standard form, we can proceed with factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. In this case, we can observe that both terms on the left side of the equation have a common factor of 2x. Factoring out 2x, we get:
2x(x - 4) = 0
This factored form of the equation is crucial because it allows us to apply the zero-product property. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if A * B = 0, then either A = 0 or B = 0 (or both). Applying this property to our equation, we set each factor equal to zero:
2x = 0 or x - 4 = 0
Now, we solve each of these linear equations separately. For 2x = 0, we divide both sides by 2 to get:
x = 0
For x - 4 = 0, we add 4 to both sides to get:
x = 4
Thus, the solutions to the equation 2x² = 8x are x = 0 and x = 4. These values form the truth set of the equation, meaning they are the values of x that make the equation true.
The Truth Set and Its Significance
The truth set, also known as the solution set, is the set of all values that satisfy a given equation. In the context of the quadratic equation 2x² = 8x, the truth set is {0, 4}. This means that substituting either 0 or 4 for x in the original equation will result in a true statement. For instance, if we substitute x = 0, we get 2(0)² = 8(0), which simplifies to 0 = 0, a true statement. Similarly, if we substitute x = 4, we get 2(4)² = 8(4), which simplifies to 32 = 32, another true statement.
The truth set provides a complete and concise representation of the solutions to an equation. It is a fundamental concept in mathematics, particularly in algebra and calculus, where solving equations and inequalities is a common task. Understanding the truth set helps in analyzing the behavior of functions, finding intercepts, and solving real-world problems that can be modeled using equations.
Alternative Methods for Solving Quadratic Equations
While factoring is an efficient method for solving 2x² = 8x, it is not always the most suitable approach for all quadratic equations. Other methods, such as completing the square and using the quadratic formula, offer alternative pathways to finding the solutions. Completing the square involves manipulating the quadratic equation to form a perfect square trinomial, which can then be easily solved by taking the square root. This method is particularly useful when the quadratic expression is not easily factorable.
The quadratic formula is a general formula that provides the solutions to any quadratic equation in the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
The quadratic formula is a powerful tool because it can be applied to any quadratic equation, regardless of whether it is factorable or not. It is especially useful when dealing with equations that have irrational or complex solutions. In the case of 2x² = 8x, we can rewrite it as 2x² - 8x = 0, where a = 2, b = -8, and c = 0. Substituting these values into the quadratic formula, we get:
x = (8 ± √((-8)² - 4 * 2 * 0)) / (2 * 2)
x = (8 ± √(64)) / 4
x = (8 ± 8) / 4
This gives us two solutions:
x = (8 + 8) / 4 = 16 / 4 = 4
x = (8 - 8) / 4 = 0 / 4 = 0
As we can see, the quadratic formula yields the same solutions as factoring, x = 0 and x = 4. Understanding and mastering these different methods for solving quadratic equations provides a comprehensive toolkit for tackling various mathematical problems.
The Importance of Quadratic Equations
Quadratic equations are not just abstract mathematical concepts; they have numerous applications in various fields, including physics, engineering, economics, and computer science. In physics, quadratic equations are used to describe projectile motion, the trajectory of objects under the influence of gravity. For example, the height of a ball thrown into the air can be modeled using a quadratic equation, allowing us to determine the maximum height the ball reaches and the time it takes to hit the ground.
In engineering, quadratic equations are used in designing structures, analyzing circuits, and optimizing processes. For instance, the shape of a parabolic arch in a bridge can be described using a quadratic equation, ensuring structural stability and efficient load distribution. In economics, quadratic equations can model supply and demand curves, helping to determine equilibrium prices and quantities. In computer science, quadratic equations are used in algorithms for optimization and machine learning.
Moreover, understanding quadratic equations is essential for further studies in mathematics. They form the foundation for more advanced topics such as calculus, differential equations, and linear algebra. The ability to solve quadratic equations is a prerequisite for mastering these subjects and applying them to real-world problems. Therefore, a solid understanding of quadratic equations is not only valuable in itself but also serves as a stepping stone for broader mathematical proficiency.
Conclusion
In conclusion, finding the truth set of the equation 2x² = 8x involves rearranging the equation into standard form, factoring out common factors, and applying the zero-product property. The solutions, x = 0 and x = 4, form the truth set {0, 4}, representing the values that satisfy the equation. While factoring is an efficient method for this particular equation, alternative methods like completing the square and using the quadratic formula provide versatile tools for solving a wide range of quadratic equations. Understanding quadratic equations is crucial due to their numerous applications in various fields and their foundational role in advanced mathematical studies. By mastering the techniques for solving quadratic equations, we equip ourselves with valuable problem-solving skills applicable to both theoretical and practical contexts. The ability to manipulate and solve these equations is a cornerstone of mathematical literacy and a key to unlocking further mathematical insights.