Solving The Quadratic Equation 2x² - 4x + 2 = 0 A Step-by-Step Guide

by Henrik Larsen 69 views

Hey guys! Today, we're diving into the exciting world of quadratic equations and tackling a specific problem. We need to find the solution set for the equation 2x² - 4x + 2 = 0 within the realm of real numbers (R). The options we have are:

A) {1} B) {0} C) {2} D) {1, 2}

And of course, we'll break down how to arrive at the correct answer, making sure everyone understands the process. So, let's put on our math hats and get started!

Understanding Quadratic Equations

Before we jump into solving this specific equation, let's take a step back and understand what quadratic equations are all about. In essence, quadratic equations are polynomial equations of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it would become a linear equation). The coefficients 'a', 'b', and 'c' play crucial roles in determining the nature and values of the solutions (also called roots) of the equation. The coefficient 'a' dictates the parabola's concavity when the quadratic equation is graphically represented; 'b' influences the axis of symmetry; and 'c' determines the y-intercept. Understanding these coefficients is crucial for choosing the appropriate method to solve the equation and interpreting the solutions within the context of the problem. Moreover, quadratic equations frequently appear in various mathematical and real-world contexts, including physics (projectile motion), engineering (designing parabolic structures), and computer science (optimization algorithms), highlighting their wide applicability and importance. Mastery of quadratic equations not only enhances problem-solving skills in mathematics but also provides a solid foundation for advanced studies in various scientific and technical fields. For instance, in physics, understanding the roots of a quadratic equation can help determine the time at which a projectile hits the ground or the points at which the potential energy of a system is minimized. In engineering, quadratic equations are used to model various systems, such as electrical circuits, mechanical vibrations, and fluid dynamics. This broad applicability makes the study of quadratic equations a cornerstone of mathematical education. The solutions to a quadratic equation represent the x-intercepts of the parabola defined by the equation, which are the points where the parabola crosses the x-axis. These solutions can be real or complex numbers, depending on the discriminant, which we'll discuss later. Solving quadratic equations effectively requires a combination of algebraic manipulation, pattern recognition, and careful attention to detail. The ability to confidently tackle quadratic equations opens doors to more advanced mathematical concepts and practical applications, solidifying its status as a fundamental skill in the quantitative disciplines.

Methods to Solve Quadratic Equations

Now that we have a handle on what quadratic equations are, let's explore the common methods we use to solve them. There are typically three main approaches:

  1. Factoring: This method involves breaking down the quadratic expression into two linear factors. If we can rewrite the equation in the form (x - r₁)(x - r₂) = 0, then the solutions are simply x = r₁ and x = r₂. Factoring is often the quickest method, but it's not always straightforward, especially when the coefficients are large or the roots are not integers.
  2. Quadratic Formula: This is the go-to method when factoring seems difficult or impossible. The quadratic formula provides a direct way to find the solutions, regardless of the complexity of the equation. For a quadratic equation in the form ax² + bx + c = 0, the quadratic formula states that the solutions are given by: x = (-b ± √(b² - 4ac)) / 2a. This formula is derived from the method of completing the square and guarantees a solution for any quadratic equation, whether the roots are real or complex. The quadratic formula is a powerful tool because it is universally applicable and requires only the identification of the coefficients a, b, and c. Its systematic approach eliminates the guesswork sometimes involved in factoring and ensures accurate solutions, provided the formula is applied correctly. The term inside the square root, (b² - 4ac), known as the discriminant, plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it's zero, the equation has one real root (a repeated root); and if it's negative, the equation has two complex roots. Understanding the discriminant provides valuable insight into the solution set without having to fully solve the equation. Moreover, the quadratic formula is essential in various applications, such as determining the points of intersection between a parabola and a line or solving optimization problems in calculus. Its versatility and reliability make it an indispensable tool in mathematics and related fields.
  3. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. It's a bit more involved than factoring but serves as the foundation for deriving the quadratic formula. The process of completing the square involves adding and subtracting (b/2a)² to the equation ax² + bx + c = 0, which allows the quadratic expression to be rewritten as a squared term plus a constant. This method is particularly useful in understanding the structure of quadratic equations and the relationship between the coefficients and the roots. Completing the square can also be applied to solve various problems beyond finding roots, such as converting the general form of a circle's equation to its standard form or finding the vertex of a parabola. While the quadratic formula is often preferred for its directness, mastering the technique of completing the square enhances one's understanding of algebraic manipulation and provides a deeper appreciation of the underlying principles of quadratic equations. Furthermore, the skill of completing the square is transferable to other areas of mathematics, such as calculus and differential equations, where similar techniques are used to solve more complex problems. Its conceptual importance and versatility make it a valuable tool in the mathematician's arsenal.

Solving the Equation 2x² - 4x + 2 = 0

Alright, let's get back to our original problem: 2x² - 4x + 2 = 0. We need to find the solution set within the real numbers.

First, notice that all the coefficients (2, -4, and 2) are divisible by 2. This means we can simplify the equation by dividing both sides by 2:

x² - 2x + 1 = 0

This simplified equation is much easier to work with. Now, let's try factoring. Can we find two numbers that multiply to 1 and add up to -2? You bet! The numbers are -1 and -1. So, we can factor the equation as:

(x - 1)(x - 1) = 0

Or, more simply:

(x - 1)² = 0

This tells us that the only solution is when x - 1 = 0, which means x = 1. So, the solution set is {1}.

We could also have used the quadratic formula. In the simplified equation x² - 2x + 1 = 0, we have a = 1, b = -2, and c = 1. Plugging these values into the quadratic formula:

x = (-(-2) ± √((-2)² - 4 * 1 * 1)) / (2 * 1) x = (2 ± √(4 - 4)) / 2 x = (2 ± √0) / 2 x = 2 / 2 x = 1

Again, we arrive at the same solution: x = 1. The quadratic formula confirms our factoring result, showing that there is exactly one real root.

Why Factoring Works So Well Here

In this specific case, factoring was a particularly efficient method due to the nature of the equation. Factoring relies on recognizing patterns and relationships between the coefficients of the quadratic expression. When the quadratic expression is a perfect square trinomial, as in our simplified equation x² - 2x + 1 = 0, factoring becomes straightforward. A perfect square trinomial is a quadratic expression that can be written in the form (ax + b)² or (ax - b)². Recognizing this pattern allows us to quickly rewrite the quadratic equation in its factored form, making it easy to identify the roots. In our example, x² - 2x + 1 fits the pattern (x - b)², where b = 1. This made it immediately apparent that the equation could be factored as (x - 1)² = 0, leading to the single root x = 1. Factoring is especially advantageous when the roots are integers or simple fractions because these roots often correspond to integer factors of the constant term. However, not all quadratic equations are easily factorable. When the coefficients are large, the roots are irrational, or the expression doesn't fit a recognizable pattern, other methods like the quadratic formula become more practical. The ability to quickly assess whether factoring is a viable approach and to efficiently execute the factoring process is a valuable skill in solving quadratic equations. It not only saves time but also enhances understanding of the underlying algebraic relationships. Mastering factoring, therefore, provides a solid foundation for tackling more complex mathematical problems and is an essential tool in any mathematician's toolkit.

The Discriminant: A Sneak Peek at the Solutions

While we successfully solved the equation using factoring and the quadratic formula, there's another valuable tool we can use to understand the nature of the solutions even before we fully solve the equation: the discriminant. The discriminant is the part of the quadratic formula under the square root sign: b² - 4ac. This simple expression tells us a lot about the roots of the quadratic equation.

  • If b² - 4ac > 0, the equation has two distinct real roots.
  • If b² - 4ac = 0, the equation has one real root (a repeated root).
  • If b² - 4ac < 0, the equation has two complex roots.

For our simplified equation, x² - 2x + 1 = 0, we have a = 1, b = -2, and c = 1. Let's calculate the discriminant:

b² - 4ac = (-2)² - 4 * 1 * 1 = 4 - 4 = 0

The discriminant is 0, which confirms that our equation has one real root, just as we found earlier. The discriminant is a powerful analytical tool because it provides immediate insight into the type and number of solutions a quadratic equation possesses, without the need for complete solution. This is particularly useful in situations where understanding the nature of the roots is more important than finding their exact values. For instance, in physics, the discriminant can help determine whether a projectile trajectory intersects a certain height or whether a system has stable equilibrium points. In engineering, it can be used to assess the stability of control systems or the feasibility of a design. The ability to quickly calculate and interpret the discriminant saves time and effort, guiding the problem-solving process and ensuring that the appropriate methods are applied. Furthermore, the discriminant connects the algebraic properties of a quadratic equation to its graphical representation. A positive discriminant corresponds to a parabola that intersects the x-axis at two distinct points, a zero discriminant to a parabola that touches the x-axis at one point, and a negative discriminant to a parabola that does not intersect the x-axis. This visual interpretation enhances understanding and provides a comprehensive perspective on quadratic equations and their applications.

Conclusion: The Correct Answer and Why

So, after simplifying the equation 2x² - 4x + 2 = 0 to x² - 2x + 1 = 0, we found that the solution set is {1}. This means the correct answer is A) {1}. We arrived at this answer through factoring and confirmed it using the quadratic formula and the discriminant.

Remember, guys, understanding the underlying concepts and methods is key to solving these types of problems. Practice makes perfect, so keep working on those quadratic equations!

Hopefully, this explanation has cleared things up for you. If you have any more questions, feel free to ask. Happy problem-solving!