Solving K² - K - 11 = 0 A Comprehensive Guide
In mathematics, solving equations is a fundamental skill. Among various types of equations, quadratic equations hold a significant place due to their wide applications in various fields like physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree. 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 are also known as its roots. These roots can be real or complex numbers.
This article will delve into the process of solving a specific quadratic equation: k² - k - 11 = 0. We will explore different methods to find the solutions for k, including factoring, completing the square, and the quadratic formula. The goal is to fully simplify the solutions and present them in a clear and understandable manner. Understanding how to solve quadratic equations is crucial for various mathematical and real-world problems. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide a comprehensive approach to solving quadratic equations.
Before we dive into solving the equation k² - k - 11 = 0, it's crucial to grasp the fundamental concepts of quadratic equations. A quadratic equation is a polynomial equation of the second degree, which means the highest power of the variable in the equation is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The coefficients a, b, and c play a significant role in determining the nature and values of the solutions, also known as roots, of the equation.
In the equation k² - k - 11 = 0, we can identify the coefficients as follows: a = 1, b = -1, and c = -11. Understanding these coefficients is the first step in choosing an appropriate method to solve the equation. The roots of a quadratic equation are the values of the variable (in this case, k) that satisfy the equation, making the left-hand side equal to zero. These roots represent the points where the parabola, which is the graphical representation of the quadratic equation, intersects the x-axis.
There are several methods to solve quadratic equations, each with its advantages and disadvantages. Factoring is a straightforward method when the equation can be easily factored. Completing the square is a method that transforms the equation into a perfect square trinomial, making it easier to solve. The quadratic formula is a general method that can be applied to any quadratic equation, regardless of whether it can be factored or not. The discriminant, which is a part of the quadratic formula (b² - 4ac), provides valuable information about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Understanding these concepts will help us choose the most efficient method to solve k² - k - 11 = 0 and interpret the results.
There are several methods available to solve quadratic equations, each suited to different situations and offering unique advantages. The three primary methods we'll explore are factoring, completing the square, and the quadratic formula. Each method provides a different approach to finding the roots of a quadratic equation, and understanding them will equip you with a versatile toolkit for solving such equations.
1. Factoring
Factoring is a method that involves expressing the quadratic equation as a product of two binomials. This method is efficient when the quadratic expression can be easily factored. For instance, if we have an equation like x² + 5x + 6 = 0, we can factor it as (x + 2)(x + 3) = 0. Setting each factor to zero gives us the solutions x = -2 and x = -3. However, not all quadratic equations can be easily factored, especially when the roots are not integers or simple fractions. In such cases, alternative methods like completing the square or the quadratic formula are more appropriate.
2. Completing the Square
Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. This method involves manipulating the equation to create a square on one side. For example, to solve x² + 6x + 5 = 0 by completing the square, we first rewrite it as x² + 6x = -5. Then, we add (6/2)² = 9 to both sides to complete the square: x² + 6x + 9 = -5 + 9, which simplifies to (x + 3)² = 4. Taking the square root of both sides gives x + 3 = ±2, leading to the solutions x = -1 and x = -5. Completing the square is a powerful method, but it can be more complex than factoring for some equations.
3. Quadratic Formula
The quadratic formula is a general solution that can be used to solve any quadratic equation of the form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). This formula is derived from the method of completing the square and provides a direct way to find the roots of the equation. The term inside the square root, b² - 4ac, is known as the discriminant, and it determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. The quadratic formula is particularly useful when the equation cannot be easily factored or when the coefficients are such that completing the square is cumbersome. In the case of k² - k - 11 = 0, we will find that the quadratic formula is the most efficient method to find the solutions.
Given the quadratic equation k² - k - 11 = 0, we can identify the coefficients as a = 1, b = -1, and c = -11. Since this equation does not factor easily, and completing the square might be cumbersome, the most efficient method to find the solutions for k is the quadratic formula. The quadratic formula is given by:
k = (-b ± √(b² - 4ac)) / (2a)
Now, let's substitute the values of a, b, and c into the formula:
k = (-(-1) ± √((-1)² - 4(1)(-11))) / (2(1))
First, simplify the expression inside the square root:
(-1)² - 4(1)(-11) = 1 + 44 = 45
So, the equation becomes:
k = (1 ± √45) / 2
Now, we need to simplify the square root of 45. We can factor 45 as 9 × 5, where 9 is a perfect square. Therefore, √45 = √(9 × 5) = √9 × √5 = 3√5. Substituting this back into the equation, we get:
k = (1 ± 3√5) / 2
This gives us two distinct solutions for k:
- k₁ = (1 + 3√5) / 2
- k₂ = (1 - 3√5) / 2
These are the fully simplified solutions for the equation k² - k - 11 = 0. The solutions are irrational numbers due to the presence of the square root of 5. The quadratic formula provides a straightforward way to find these solutions, especially when other methods like factoring are not practical. The discriminant (b² - 4ac) in this case is 45, which is positive, indicating that we have two distinct real roots, as we have found.
After applying the quadratic formula to the equation k² - k - 11 = 0, we obtained the solutions k = (1 ± 3√5) / 2. These solutions are already in a simplified form, but it's essential to understand what full simplification means in this context. The solutions involve a radical term (√5), which cannot be further simplified as 5 is a prime number and has no perfect square factors other than 1. The expression is also a fraction, and we should ensure that the numerator and denominator have no common factors that can be canceled out.
The solutions are presented as two separate values:
- k₁ = (1 + 3√5) / 2
- k₂ = (1 - 3√5) / 2
Each solution is a combination of a rational number (1/2) and an irrational number ((3√5)/2). These numbers cannot be combined further because they are of different types. The solutions are exact values, and while they can be approximated using a calculator, the exact form provides a more precise representation of the roots of the equation. In many mathematical contexts, especially in higher-level mathematics, exact solutions are preferred over decimal approximations.
The solutions are also in the simplest form concerning the denominator. The denominator, 2, is a prime number, and neither the rational part (1) nor the irrational part (3√5) has any factors that can cancel out the 2. Therefore, the solutions are fully simplified. In summary, simplifying solutions often involves reducing radicals, canceling common factors in fractions, and ensuring that the expression is in its most concise form. In this case, the solutions k₁ = (1 + 3√5) / 2 and k₂ = (1 - 3√5) / 2 are already in their simplest form and represent the exact roots of the quadratic equation k² - k - 11 = 0.
In this comprehensive guide, we have successfully solved the quadratic equation k² - k - 11 = 0 and fully simplified the solutions. We began by understanding the fundamentals of quadratic equations, recognizing the standard form ax² + bx + c = 0, and identifying the coefficients in our specific equation. We then explored various methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. Recognizing that factoring was not straightforward and completing the square could be cumbersome, we opted for the quadratic formula as the most efficient method.
Applying the quadratic formula, k = (-b ± √(b² - 4ac)) / (2a), we substituted the coefficients a = 1, b = -1, and c = -11. This led us to the solutions k = (1 ± √45) / 2. We further simplified √45 as 3√5, resulting in the solutions k₁ = (1 + 3√5) / 2 and k₂ = (1 - 3√5) / 2. These solutions are the exact roots of the equation and are in their simplest form. The presence of the radical term √5 indicates that the roots are irrational numbers.
Understanding how to solve quadratic equations is a crucial skill in mathematics, with applications spanning various fields. The quadratic formula, in particular, is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity. By following the steps outlined in this guide, you can confidently solve quadratic equations and simplify the solutions. The ability to solve such equations is not only valuable in academic settings but also in practical situations where mathematical modeling is required.
