Solving 6x² - X - 1 = 0 Methods And Solutions Explained

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Finding the solutions to the quadratic equation 6x² - x - 1 = 0 is a fundamental problem in algebra. Quadratic equations, which take the general form ax² + bx + c = 0, where a, b, and c are constants, appear in various fields, including physics, engineering, and economics. Mastering the techniques to solve them is crucial for anyone delving into these disciplines. This article provides a detailed exploration of the methods to solve this specific equation, offering insights and steps for a clear understanding. We will discuss factoring, the quadratic formula, and completing the square, offering a comprehensive approach to tackling quadratic equations. Solving 6x² - x - 1 = 0 isn't just about finding answers; it's about understanding the underlying principles that govern quadratic equations and their applications in the real world. The ability to solve such equations opens doors to modeling and solving numerous real-world problems. The solutions, also known as roots or zeros, represent the points where the parabola defined by the equation intersects the x-axis. These points are critical in understanding the behavior of the quadratic function and its applications in various fields. Therefore, a thorough understanding of these methods is essential for anyone seeking to deepen their mathematical knowledge and problem-solving skills.

Method 1: Factoring

Factoring is often the quickest and most intuitive method for solving quadratic equations, provided the equation can be factored easily. The core idea behind factoring is to express the quadratic equation as a product of two binomials. This approach relies on reversing the process of expanding two binomials using the distributive property (also known as FOIL – First, Outer, Inner, Last). For the equation 6x² - x - 1 = 0, we aim to find two binomials of the form (px + q) and (rx + s) such that their product equals the given quadratic expression. To factor 6x² - x - 1 = 0, we need to find two numbers that multiply to give the product of the leading coefficient (6) and the constant term (-1), which is -6, and add up to the middle coefficient (-1). The numbers -3 and 2 satisfy these conditions because (-3) * 2 = -6 and (-3) + 2 = -1. We can then rewrite the middle term (-x) as -3x + 2x, allowing us to factor by grouping. This process transforms the original quadratic equation into a more manageable form, making it easier to identify the roots. Once the equation is factored, we can set each factor equal to zero and solve for x. This method not only provides the solutions but also enhances our understanding of the relationship between the coefficients and the roots of the quadratic equation. Factoring is a powerful tool in algebra, and mastering it can significantly simplify the process of solving quadratic equations. By understanding the principles behind factoring, we can efficiently tackle a wide range of quadratic equations, making it an essential skill for anyone studying mathematics.

Step-by-Step Factoring

  1. Rewrite the middle term: Replace -x with -3x + 2x, so the equation becomes 6x² - 3x + 2x - 1 = 0.
  2. Factor by grouping:
    • From the first two terms, 6x² - 3x, factor out the greatest common factor (GCF), which is 3x. This gives us 3x(2x - 1).
    • From the last two terms, 2x - 1, the GCF is 1, so we have 1(2x - 1).
    • Now the equation looks like: 3x(2x - 1) + 1(2x - 1) = 0.
  3. Factor out the common binomial: Notice that (2x - 1) is common to both terms. Factor it out to get (2x - 1)(3x + 1) = 0.
  4. Set each factor to zero:
    • 2x - 1 = 0 => 2x = 1 => x = 1/2
    • 3x + 1 = 0 => 3x = -1 => x = -1/3

Therefore, the solutions obtained by factoring are x = 1/2 and x = -1/3. These solutions are the values of x that make the equation 6x² - x - 1 = 0 true. Factoring is not just a method for finding solutions; it's a way of understanding the structure of the equation and how the roots relate to the coefficients. By mastering the factoring technique, one can develop a deeper understanding of algebraic manipulation and problem-solving. This step-by-step approach provides a clear and concise way to factor the given quadratic equation, highlighting the importance of identifying common factors and applying the distributive property in reverse. The solutions obtained through factoring are crucial for various applications in mathematics and other fields, emphasizing the significance of this method in solving quadratic equations.

Method 2: Quadratic Formula

The quadratic formula is a universally applicable method for solving any quadratic equation, regardless of whether it can be factored easily. This formula provides a direct pathway to the solutions, making it an indispensable tool in algebra. The quadratic formula is derived from the process of completing the square and is given by:

x = [-b ± √(b² - 4ac)] / (2a)

where a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. This formula is particularly useful when factoring is difficult or impossible, as it provides a reliable method to find the roots of any quadratic equation. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant gives us 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 one real solution (a repeated root); and if it is negative, there are two complex solutions. For the equation 6x² - x - 1 = 0, applying the quadratic formula involves identifying the coefficients a, b, and c, substituting them into the formula, and simplifying the expression. This process yields the solutions for x, which are the values that satisfy the quadratic equation. The quadratic formula is not just a computational tool; it's a fundamental concept in algebra that connects the coefficients of a quadratic equation to its roots. Understanding and applying this formula is crucial for solving a wide range of mathematical problems and for building a strong foundation in algebra.

Applying the Quadratic Formula

  1. Identify coefficients: In the equation 6x² - x - 1 = 0, we have a = 6, b = -1, and c = -1.
  2. Substitute into the formula:
    • x = [-(-1) ± √((-1)² - 4 * 6 * (-1))] / (2 * 6)
    • x = [1 ± √(1 + 24)] / 12
    • x = [1 ± √25] / 12
    • x = [1 ± 5] / 12
  3. Calculate the two solutions:
    • x₁ = (1 + 5) / 12 = 6 / 12 = 1/2
    • x₂ = (1 - 5) / 12 = -4 / 12 = -1/3

Thus, using the quadratic formula, we find the solutions to be x = 1/2 and x = -1/3, which match the solutions obtained through factoring. This consistency validates the accuracy of both methods. The quadratic formula's ability to handle any quadratic equation makes it an essential tool in mathematics. Its application ensures that solutions can be found even when factoring is not feasible. This step-by-step application of the quadratic formula showcases its power and versatility in solving quadratic equations. The formula not only provides solutions but also deepens our understanding of the relationship between the coefficients and the roots of the equation. The ability to apply this formula efficiently is a key skill for anyone studying algebra and related fields.

Method 3: Completing the Square

Completing the square is another powerful technique for solving quadratic equations. This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve for x. While it can be slightly more involved than factoring or using the quadratic formula, completing the square is a valuable skill, especially for understanding the derivation of the quadratic formula itself. Completing the square involves manipulating the equation to create a perfect square trinomial on one side, which can then be factored into the form (x + p)² or (x - p)². This transformation allows us to isolate x by taking the square root of both sides of the equation. For the equation 6x² - x - 1 = 0, we first need to ensure that the coefficient of the x² term is 1. If it's not, we divide the entire equation by that coefficient. Then, we move the constant term to the right side of the equation and add a value to both sides that completes the square. This value is determined by taking half of the coefficient of the x term, squaring it, and adding it to both sides. This process creates a perfect square trinomial on the left side, which can be easily factored. The method of completing the square not only solves quadratic equations but also provides insights into the structure and properties of quadratic expressions. It's a fundamental technique in algebra that enhances problem-solving skills and deepens mathematical understanding. By mastering completing the square, one gains a more comprehensive grasp of quadratic equations and their solutions.

Step-by-Step Completing the Square

  1. Divide by the leading coefficient: Divide the entire equation 6x² - x - 1 = 0 by 6 to make the coefficient of x² equal to 1:
    • x² - (1/6)x - 1/6 = 0
  2. Move the constant term to the right side:
    • x² - (1/6)x = 1/6
  3. Complete the square:
    • Take half of the coefficient of the x term, which is (-1/6) / 2 = -1/12. Square this value: (-1/12)² = 1/144.
    • Add this to both sides of the equation:
      • x² - (1/6)x + 1/144 = 1/6 + 1/144
  4. Factor the left side as a perfect square trinomial:
    • (x - 1/12)² = 25/144
  5. Take the square root of both sides:
    • x - 1/12 = ±√(25/144)
    • x - 1/12 = ±5/12
  6. Solve for x:
    • x = 1/12 ± 5/12
    • x₁ = (1/12 + 5/12) = 6/12 = 1/2
    • x₂ = (1/12 - 5/12) = -4/12 = -1/3

Again, we find the solutions to be x = 1/2 and x = -1/3, consistent with the other methods. Completing the square is a robust method that provides a clear path to the solutions, even when factoring is not straightforward. This method's importance extends beyond solving equations, as it is a cornerstone in the derivation of the quadratic formula and in understanding the transformations of quadratic functions. The step-by-step breakdown illustrates the process of transforming a quadratic equation into a form that allows for easy isolation of the variable, reinforcing the fundamental algebraic principles involved. The consistency of the solutions obtained through completing the square with those found using other methods underscores its reliability and the interconnectedness of algebraic techniques.

Conclusion

In summary, we have explored three distinct methods for solving the quadratic equation 6x² - x - 1 = 0: factoring, using the quadratic formula, and completing the square. Each method provides a unique approach to finding the solutions, and each has its advantages and applications. Factoring is often the quickest method when the quadratic expression can be easily factored, as it relies on reversing the distributive property. The quadratic formula, on the other hand, is a universally applicable method that works for any quadratic equation, making it an essential tool for solving complex problems. Completing the square, while sometimes more involved, offers a deeper understanding of the structure of quadratic equations and is crucial for the derivation of the quadratic formula. The solutions obtained through all three methods – x = 1/2 and x = -1/3 – are consistent, demonstrating the reliability and accuracy of these techniques. Mastering these methods is essential for anyone studying mathematics, as quadratic equations appear in various contexts, from basic algebra to advanced calculus and physics. Understanding how to solve them not only provides solutions to specific problems but also enhances overall problem-solving skills and mathematical intuition. The ability to choose the most appropriate method for a given problem is a key aspect of mathematical proficiency, and this article has aimed to provide a comprehensive understanding of the tools available for solving quadratic equations. Whether through factoring, the quadratic formula, or completing the square, the goal is to find the values of x that satisfy the equation, thereby unlocking a deeper understanding of the mathematical relationships at play.