For Which Value Of Is

Solving for x: A Deep Dive into the Equation x² - 6x + 8 = 0

This article explores the solution to the quadratic equation x² - 6x + 8 = 0. We'll cover several methods to solve this equation, emphasizing the underlying mathematical principles and demonstrating different approaches to arrive at the same solution. Understanding how to solve quadratic equations is fundamental in algebra and has wide-ranging applications in various fields, from physics and engineering to economics and computer science. This comprehensive guide will equip you with the knowledge and skills to tackle similar problems effectively.

Meta Description: Learn how to solve the quadratic equation x² - 6x + 8 = 0 using various methods, including factoring, the quadratic formula, and completing the square. This detailed guide provides step-by-step solutions and explains the underlying mathematical concepts.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our specific equation, x² - 6x + 8 = 0, fits this form with a = 1, b = -6, and c = 8.

Solving a quadratic equation means finding the values of 'x' that make the equation true. These values are called the roots, solutions, or zeros of the equation. A quadratic equation can have up to two real roots, one real root (a repeated root), or two complex roots.

Method 1: Factoring

Factoring is often the quickest and easiest method to solve a quadratic equation, especially when the equation is easily factorable. The goal is to rewrite the quadratic expression as a product of two linear expressions.

Let's factor x² - 6x + 8 = 0:

We look for two numbers that add up to -6 (the coefficient of x) and multiply to 8 (the constant term). These numbers are -2 and -4. Therefore, we can factor the equation as:

(x - 2)(x - 4) = 0

This equation is true if either (x - 2) = 0 or (x - 4) = 0. Solving these linear equations gives us:

x - 2 = 0 => x = 2 x - 4 = 0 => x = 4

Therefore, the solutions to the equation x² - 6x + 8 = 0 are x = 2 and x = 4.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it's easily factorable or not. The formula is derived from completing the square (explained in the next section) and provides a direct way to calculate the roots.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For our equation, x² - 6x + 8 = 0, we have a = 1, b = -6, and c = 8. Substituting these values into the quadratic formula, we get:

x = [-(-6) ± √((-6)² - 4 * 1 * 8)] / (2 * 1) x = [6 ± √(36 - 32)] / 2 x = [6 ± √4] / 2 x = [6 ± 2] / 2

This gives us two solutions:

x = (6 + 2) / 2 = 8 / 2 = 4 x = (6 - 2) / 2 = 4 / 2 = 2

Again, the solutions are x = 2 and x = 4.

Method 3: Completing the Square

Completing the square is a technique used to manipulate a quadratic expression into a perfect square trinomial, which can then be easily factored. It's a more involved method than factoring but provides a valuable understanding of the underlying structure of quadratic equations.

To complete the square for x² - 6x + 8 = 0, we follow these steps:

1. Move the constant term to the right side: x² - 6x = -8

2. Take half of the coefficient of x, square it, and add it to both sides: The coefficient of x is -6. Half of -6 is -3, and (-3)² = 9. Adding 9 to both sides: x² - 6x + 9 = -8 + 9 x² - 6x + 9 = 1

3. Factor the left side as a perfect square trinomial: (x - 3)² = 1

4. Take the square root of both sides: x - 3 = ±√1 x - 3 = ±1

5. Solve for x: x = 3 + 1 = 4 x = 3 - 1 = 2

Once again, we arrive at the solutions x = 2 and x = 4.

Understanding the Discriminant

The discriminant (b² - 4ac) within the quadratic formula provides valuable information about the nature of the roots of a quadratic equation.

• If b² - 4ac > 0: The equation has two distinct real roots. This is the case for our equation (36 - 32 = 4 > 0).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots (involving imaginary numbers).

Applications of Quadratic Equations

Quadratic equations have numerous applications across various disciplines:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stress and strain in materials.
• Economics: Modeling supply and demand, analyzing cost functions.
• Computer Science: Developing algorithms, solving optimization problems.
• Mathematics: Solving geometric problems, understanding conic sections.

Further Exploration

This article provides a solid foundation for understanding and solving quadratic equations. Further exploration could involve:

• Solving more complex quadratic equations: Equations with fractional or decimal coefficients, or equations requiring the use of more advanced techniques.
• Exploring the relationship between the roots and coefficients of a quadratic equation: Vieta's formulas provide valuable insights into this relationship.
• Investigating cubic and higher-degree polynomial equations: These equations represent more complex mathematical relationships and require different solution methods.
• Applying quadratic equations to real-world problems: Working through practical examples can solidify your understanding and develop problem-solving skills.

By mastering the techniques presented in this article, you'll be well-equipped to tackle a wide range of quadratic equation problems and appreciate their significance in various fields of study. Remember to practice regularly to build confidence and proficiency in solving these fundamental equations. The key is understanding the underlying principles, choosing the appropriate method, and carefully executing the steps. With practice, solving quadratic equations will become second nature.