Solving (2x+3)/(x+3) = X A Step-by-Step Algebraic Analysis

Jul 12, 2025 by ADMIN 59 views

Can (2x+3)/(x+3) = x Be Reduced to x? A Detailed Analysis

Introduction

The equation (2x+3)/(x+3) = x presents an interesting problem in algebra. At first glance, it might seem like a straightforward equation to solve, but a closer look reveals the nuances involved in simplifying and finding the correct solutions. In this comprehensive analysis, we will delve into the steps required to solve the equation, examine potential pitfalls, and determine whether the equation can indeed be reduced such that the answer is simply x. This exploration is crucial for students, educators, and anyone keen on enhancing their algebraic problem-solving skills. Understanding the underlying principles will help in tackling similar equations with confidence and precision.

Solving the Equation (2x+3)/(x+3) = x

To determine if the equation ${ rac{2x+3}{x+3} = x}$ can be reduced to x, we need to solve it step by step. The initial step involves eliminating the fraction, which is a common technique in solving algebraic equations. We achieve this by multiplying both sides of the equation by the denominator, which in this case is (x+3). This process helps in transforming the equation into a more manageable form, allowing us to proceed with further simplification. It’s a crucial step because it clears the fraction and sets the stage for rearranging the terms into a standard polynomial form. However, we must also be cautious about potential extraneous solutions that might arise due to this multiplication.

Step-by-Step Solution

Multiply both sides by (x+3): ${(x+3) rac{2x+3}{x+3} = x(x+3)}$ This step is pivotal as it removes the fraction, making the equation easier to handle. The multiplication on the left side cancels out the denominator, while on the right side, x is distributed across (x+3). This transformation is essential for bringing all terms to one side and eventually solving for x.
Simplify: ${2x + 3 = x^2 + 3x}$ Here, we simplify both sides of the equation. The left side remains 2x + 3, while the right side expands to x^2 + 3x. This simplification is critical for reorganizing the equation into a quadratic form, which we can then solve using various methods such as factoring, completing the square, or using the quadratic formula.
Rearrange the equation into a quadratic form: ${0 = x^2 + 3x - 2x - 3}$ ${0 = x^2 + x - 3}$ By moving all terms to one side, we set the equation equal to zero. This rearrangement is a fundamental step in solving quadratic equations. Combining like terms, we get the standard quadratic form x^2 + x - 3 = 0. This form is essential for applying the quadratic formula or attempting to factor the equation.
Solve the quadratic equation: The quadratic equation x^2 + x - 3 = 0 does not factor easily, so we use the quadratic formula: ${x = rac{-b ext{ ± } ext{√}(b^2 - 4ac)}{2a}}$ In our equation, a = 1, b = 1, and c = -3. Plugging these values into the quadratic formula, we get: ${x = rac{-1 ext{ ± } ext{√}(1^2 - 4(1)(-3))}{2(1)}}$ This step involves substituting the coefficients of the quadratic equation into the quadratic formula. The quadratic formula is a powerful tool for finding the roots of any quadratic equation, regardless of whether it can be factored. Correct substitution is crucial for obtaining the correct solutions.
Simplify further: ${x = rac{-1 ext{ \pm } ext{\sqrt}(1 + 12)}{2}}$ ${x = rac{-1 ext{ \pm } ext{\sqrt}13}{2}}$ Simplifying the expression under the square root and the entire fraction, we arrive at the solutions for x. The simplified form provides the exact values of x that satisfy the equation. These values are irrational numbers due to the presence of the square root of 13.
The solutions are: ${x = rac{-1 + ext{\sqrt}13}{2} ext{ and } x = rac{-1 - ext{\sqrt}13}{2}}$ Thus, we obtain two distinct solutions for x. These solutions are the roots of the original equation and represent the values of x that make the equation true. It’s important to note that these are the exact solutions, and they are irrational numbers.

Checking for Extraneous Solutions

It is imperative to check for extraneous solutions when dealing with rational equations. Extraneous solutions are values that satisfy the transformed equation but not the original equation. They often arise when we multiply both sides of an equation by an expression that can be zero. In this case, we multiplied by (x+3), so we need to ensure that neither of our solutions makes the denominator (x+3) equal to zero.

Check x = (-1 + √13)/2: ${ rac{-1 + ext{\sqrt}13}{2} + 3 \neq 0}$ This solution does not make the denominator zero.
Check x = (-1 - √13)/2: ${ rac{-1 - ext{\sqrt}13}{2} + 3 \neq 0}$ This solution also does not make the denominator zero.

Since neither solution makes the denominator zero, both solutions are valid.

Can the Equation Be Reduced to x?

From our step-by-step solution, it is clear that the solutions to the equation are:

${x = rac{-1 + ext{\sqrt}13}{2} ext{ and } x = rac{-1 - ext{\sqrt}13}{2}}$

These solutions are not simply x. Therefore, the statement that the equation can be reduced so that the answer is x is false. The equation leads to two distinct irrational solutions, which are obtained through the application of the quadratic formula. The solutions highlight the importance of careful algebraic manipulation and the necessity of checking for extraneous solutions.

Conclusion

In summary, the equation (2x+3)/(x+3) = x cannot be reduced so that the answer is x. Solving the equation involves transforming it into a quadratic equation and applying the quadratic formula, which yields two distinct irrational solutions. This exercise underscores the importance of following a systematic approach to solving algebraic equations and verifying the solutions to avoid extraneous results. The detailed step-by-step solution provided here serves as a valuable guide for students and educators alike, enhancing their understanding of algebraic problem-solving techniques. The key takeaway is that while initial simplifications are crucial, a thorough solution requires careful consideration of all possible outcomes and potential pitfalls.

Therefore, the final answer is:

False