Solving X/2 = (3x+4)/2x A Step-by-Step Guide

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In the realm of mathematics, solving for unknown variables is a fundamental skill. Equations, the mathematical sentences that state the equality of two expressions, often contain variables that we aim to determine. One common type of equation is the algebraic equation, where variables are combined with constants through arithmetic operations. This article delves into the process of solving for x in the equation x/2 = (3x+4)/2x, providing a step-by-step guide and exploring the underlying concepts.

Understanding the Equation

The equation x/2 = (3x+4)/2x is a rational equation, meaning it involves fractions where the numerator and/or denominator contain variables. To solve this type of equation, our primary goal is to eliminate the fractions and transform the equation into a more manageable form, such as a linear or quadratic equation. This transformation involves algebraic manipulations that maintain the equality of both sides of the equation.

Before we embark on the solving process, it's crucial to identify any restrictions on the variable. In this equation, the denominator 2x cannot be equal to zero, as division by zero is undefined. Therefore, x ≠ 0 is a crucial constraint that we must keep in mind when interpreting our solutions. Any solution that violates this restriction is an extraneous solution and must be discarded.

Step-by-Step Solution

  1. Eliminate the Fractions:

    To eliminate the fractions, we multiply both sides of the equation by the least common denominator (LCD) of the fractions. In this case, the LCD is 2x. Multiplying both sides by 2x gives us:

    2x * (x/2) = 2x * ((3x+4)/2x)

    Simplifying both sides, we get:

    x^2 = 3x + 4

    This step transforms the rational equation into a quadratic equation, which we can solve using various techniques.

  2. Rearrange into Standard Quadratic Form:

    A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. To solve the quadratic equation, we first need to rearrange it into this standard form. Subtracting 3x and 4 from both sides of the equation, we obtain:

    x^2 - 3x - 4 = 0

    Now the equation is in standard quadratic form, with a = 1, b = -3, and c = -4.

  3. Solve the Quadratic Equation:

    There are several methods for solving quadratic equations, including:

    • Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If we can factor the quadratic expression, we can set each factor equal to zero and solve for x. In this case, the quadratic expression x^2 - 3x - 4 can be factored as (x - 4)(x + 1). Setting each factor equal to zero, we get:

      x - 4 = 0  or  x + 1 = 0

      Solving these linear equations, we find the solutions x = 4 and x = -1.

    • Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. The formula is:

      x = (-b ± √(b^2 - 4ac)) / 2a

      Plugging in the values a = 1, b = -3, and c = -4 into the quadratic formula, we get:

      x = (3 ± √((-3)^2 - 4 * 1 * -4)) / (2 * 1)

      Simplifying, we get:

      x = (3 ± √(9 + 16)) / 2

      x = (3 ± √25) / 2

      x = (3 ± 5) / 2

      This gives us two solutions:

      x = (3 + 5) / 2 = 4

      x = (3 - 5) / 2 = -1

    • Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial on one side. While effective, it can be more involved than factoring or using the quadratic formula.

  4. Check for Extraneous Solutions:

    Recall that we identified x ≠ 0 as a restriction on the variable. We must check if any of our solutions violate this restriction. In this case, neither x = 4 nor x = -1 is equal to zero, so both solutions are valid.

The Solutions

Therefore, the solutions to the equation x/2 = (3x+4)/2x are x = 4 and x = -1. This corresponds to option (C) x = 4, -1.

Key Concepts and Principles

  • Rational Equations: Equations involving fractions with variables in the numerator and/or denominator. Solving rational equations often involves eliminating fractions by multiplying both sides by the LCD.
  • Quadratic Equations: Equations of the form ax^2 + bx + c = 0. Quadratic equations can be solved by factoring, using the quadratic formula, or completing the square.
  • Extraneous Solutions: Solutions that arise during the solving process but do not satisfy the original equation. It's crucial to check for extraneous solutions, especially when dealing with rational equations or equations involving radicals.
  • Least Common Denominator (LCD): The smallest multiple that is divisible by all the denominators in a set of fractions. The LCD is used to eliminate fractions in rational equations.

Common Mistakes to Avoid

  • Forgetting Restrictions on Variables: Failing to identify and account for restrictions on variables, such as denominators that cannot be zero, can lead to extraneous solutions.
  • Incorrectly Eliminating Fractions: Multiplying only one side of the equation by the LCD or making errors in the multiplication process can lead to incorrect results.
  • Algebraic Errors: Mistakes in algebraic manipulation, such as incorrectly factoring or applying the quadratic formula, can also lead to incorrect solutions.

Real-World Applications

Solving equations like this has numerous applications in various fields, including:

  • Physics: Calculating projectile motion, electrical circuits, and other physical phenomena often involves solving algebraic equations.
  • Engineering: Designing structures, machines, and systems requires solving equations to determine optimal parameters and ensure stability.
  • Economics: Modeling economic systems and predicting market behavior often involves solving equations.
  • Computer Science: Developing algorithms and solving computational problems often relies on algebraic techniques.

Conclusion

Solving for x in the equation x/2 = (3x+4)/2x demonstrates the power of algebraic manipulation and the importance of understanding fundamental concepts. By eliminating fractions, transforming the equation into a standard form, and applying appropriate solving techniques, we successfully determined the solutions x = 4 and x = -1. Remember to always check for extraneous solutions and be mindful of restrictions on variables. This problem highlights the core principles of solving equations, a skill that is indispensable in mathematics and its applications across various disciplines. Mastering these techniques not only provides the ability to solve complex problems, but also fosters a deeper understanding of mathematical relationships and logical reasoning. Solving algebraic equations is a cornerstone of mathematical proficiency, opening doors to advanced concepts and real-world applications. The ability to manipulate equations, identify patterns, and apply appropriate strategies is a testament to the power of mathematical thinking. Therefore, consistent practice and a thorough understanding of underlying principles are key to unlocking the full potential of algebra and its role in shaping our understanding of the world around us. This comprehensive guide equips you with the knowledge and tools necessary to tackle similar problems with confidence, further solidifying your grasp of algebraic problem-solving. Remember, every equation solved is a step forward in your mathematical journey, paving the way for future discoveries and applications.