Solving The Equation √(2e^x - 5) = 5 - √(2e^x) A Step-by-Step Guide

Introduction

In this article, we will delve into solving the exponential equation ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$. This equation combines exponential functions with square roots, presenting an interesting challenge. We will explore the necessary steps and techniques to find the solution(s) for x. Our approach will involve algebraic manipulation, substitution, and careful consideration of the domain to ensure the validity of our solutions. This exploration is valuable not only for its mathematical rigor but also for its applications in various scientific and engineering fields where exponential models are prevalent. Solving such equations is a fundamental skill in advanced mathematics and is often encountered in calculus, differential equations, and other related disciplines. Understanding the nuances of solving equations with radicals and exponentials helps in building a strong foundation in mathematical problem-solving.

Understanding the Equation

The equation we aim to solve is ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$. Before diving into the solution process, it’s crucial to understand the components of the equation. We have an exponential term, e^x, combined with square roots. The square roots impose a condition that the expressions inside them must be non-negative. Specifically, we must have 2e^x - 5 ≥ 0 and 2e^x ≥ 0. The latter condition is always true since e^x is always positive. However, the first condition, 2e^x - 5 ≥ 0, implies that e^x ≥ 5/2. This condition will be important when we check our final solution(s).

Additionally, the term 5 - \sqrt{2e^x} on the right-hand side introduces another constraint. Since the square root is always non-negative, 5 - \sqrt{2e^x} must also be non-negative, as it equals the left-hand side, which involves a square root. Thus, we have 5 - \sqrt{2e^x} ≥ 0, which implies \sqrt{2e^x} ≤ 5. Squaring both sides gives 2e^x ≤ 25, or e^x ≤ 25/2. Combining this with the previous condition e^x ≥ 5/2, we find that our solution for x must satisfy 5/2 ≤ e^x ≤ 25/2. These constraints are essential for ensuring the validity of the solutions we obtain.

The exponential function e^x is strictly increasing, which means that if e^a = e^b, then a = b. This property will be useful when we solve for x after eliminating the square roots. Understanding the behavior of exponential functions and square roots is crucial for tackling such equations effectively. Recognizing these initial constraints and properties sets the stage for a systematic approach to solving the equation.

Step-by-Step Solution

Now, let's proceed with solving the equation ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$ step by step.

1. Isolate one of the square roots: The equation is already set up in a way that the square roots are somewhat isolated. We have ${\sqrt{2e^x - 5}}$ on one side and 5 - \sqrt{2e^x} on the other. This is a good starting point.

2. Square both sides: To eliminate the square roots, we square both sides of the equation. This gives us:

   (\sqrt{2e^x - 5})^2 = (*5 - \sqrt{2e^x}*)2

   This simplifies to:

   2e^x - 5 = 25 - 10*\sqrt{2e^x}* + 2e^x

3. Simplify the equation: Notice that the 2e^x terms cancel out on both sides, leaving us with:

   -5 = 25 - 10*\sqrt{2e^x}

4. Isolate the remaining square root: Rearrange the terms to isolate the square root term:

   10*\sqrt{2e^x}* = 30

5. Divide to simplify: Divide both sides by 10:

   \sqrt{2e^x} = 3

6. Square both sides again: To eliminate the remaining square root, square both sides:

   (*\sqrt{2e^x}*)2 = 3^2

   This simplifies to:

   2e^x = 9

7. Solve for e^x: Divide both sides by 2:

   e^x = 9/2

8. Solve for x: To solve for x, take the natural logarithm (ln) of both sides:

   x = ln(9/2)

Verification of the Solution

We found a potential solution x = ln(9/2). Now, we need to verify if this solution satisfies the original equation and the constraints we identified earlier. The original equation is:

${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$

Our solution is x = ln(9/2). Let's substitute this value into the equation:

e^(ln(9/2)) = 9/2

Now, substitute this into the original equation:

${\sqrt{2(9/2) - 5} = 5 - \sqrt{2(9/2)}}$

Simplify:

${\sqrt{9 - 5} = 5 - \sqrt{9}}$

${\sqrt{4} = 5 - 3}$

2 = 2

The solution satisfies the original equation. Next, we need to check the constraints we established earlier, which were 5/2 ≤ e^x ≤ 25/2. We found that e^x = 9/2. Let's check if this value falls within our constraints:

5/2 ≤ 9/2 ≤ 25/2

2. 5 ≤ 4.5 ≤ 12.5

The value 9/2 does indeed fall within the required range. Therefore, our solution x = ln(9/2) is valid.

Alternative Approaches

While the step-by-step method provides a clear path to the solution, there are alternative approaches one could take to solve the equation ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$. One such approach involves using substitution to simplify the equation. This method can sometimes make the algebraic manipulations easier to manage.

Substitution Method

1. Introduce a substitution: Let y = \sqrt{2e^x}. Then, y^2 = 2e^x. We can rewrite the term 2e^x - 5* as y^2 - 5. Now, the original equation becomes:

   ${\sqrt{y^2 - 5} = 5 - y}$

2. Square both sides: Squaring both sides of the equation, we get:

   y^2 - 5 = (5 - y)^2

3. Expand and simplify: Expanding the right side gives:

   y^2 - 5 = 25 - 10y + y^2

   The y^2 terms cancel out:

   -5 = 25 - 10y

4. Solve for y: Rearrange the terms to solve for y:

   10y = 30

   y = 3

5. Substitute back: Now, substitute back y = \sqrt{2e^x}:

   \sqrt{2e^x} = 3

6. Solve for e^x: Square both sides:

   2e^x = 9

   e^x = 9/2

7. Solve for x: Take the natural logarithm of both sides:

   x = ln(9/2)

This alternative substitution method arrives at the same solution, x = ln(9/2), but it offers a slightly different perspective on the problem. By introducing a new variable, we transformed the equation into a more manageable form. This method highlights the flexibility in problem-solving and demonstrates how different approaches can lead to the same result. The key is to choose a method that feels intuitive and helps simplify the equation effectively. As we showed earlier, this solution also satisfies all the necessary constraints, making it a valid solution.

Common Mistakes and How to Avoid Them

When solving equations like ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$, several common mistakes can occur. Identifying these pitfalls and understanding how to avoid them is crucial for arriving at the correct solution. Here, we will discuss some typical errors and provide guidance on preventing them.

1. Forgetting to check for extraneous solutions: One of the most frequent mistakes when dealing with radical equations is failing to verify the solutions. Squaring both sides of an equation can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original equation. In our case, after solving for x, we need to substitute the value back into the original equation to ensure it holds true. Forgetting this step can lead to accepting incorrect solutions.

   How to avoid: Always substitute the solutions back into the original equation and check if they satisfy it. If a solution does not satisfy the original equation, it is extraneous and should be discarded.

2. Ignoring domain restrictions: Radical expressions have domain restrictions. The expression inside a square root must be non-negative. In our equation, we have 2e^x - 5 and 2e^x inside square roots. Thus, we must ensure that 2e^x - 5 ≥ 0 and 2e^x ≥ 0. Additionally, since ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$, the term 5 - \sqrt{2e^x} must also be non-negative. Ignoring these restrictions can lead to solutions that are not valid.

   How to avoid: Before solving the equation, identify and note all domain restrictions imposed by the radical expressions. Check if the final solutions satisfy these restrictions. If they don't, they are not valid.

3. Incorrectly squaring binomials: When squaring binomials like (5 - \sqrt2e^x}*)2, it’s essential to use the correct formula (a - b)^2 = a^2 - 2ab + b^2. A common mistake is to square each term separately, which would incorrectly give 25 + 2e^x. The middle term, -10*\sqrt{2e^x, is often missed.

   How to avoid: Always use the correct formula for squaring binomials. Write out the expansion step by step to avoid errors.

4. Algebraic manipulation errors: Errors in algebraic manipulation, such as incorrectly adding, subtracting, multiplying, or dividing terms, can lead to incorrect solutions. This is particularly true when dealing with multiple steps and terms in an equation.

   How to avoid: Perform each step carefully and double-check your work. Break down complex steps into smaller, manageable parts. If possible, simplify expressions before proceeding with further steps.

5. Misunderstanding exponential properties: The properties of exponential functions and logarithms are crucial for solving equations involving e^x. A common mistake is to incorrectly apply these properties, such as taking the logarithm of a sum or difference. Only the logarithm of a product or quotient can be split using logarithmic properties.

   How to avoid: Review and understand the properties of exponential functions and logarithms. Apply these properties correctly and consistently.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving exponential and radical equations.

Real-World Applications

The ability to solve equations like ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$ extends beyond the realm of pure mathematics and finds applications in various real-world scenarios. Exponential functions and radical expressions appear in numerous scientific and engineering contexts, making the skills developed in solving such equations highly valuable.

1. Physics: Exponential functions are used extensively in physics to model processes such as radioactive decay, capacitor charging and discharging, and the cooling of objects. For instance, the decay of a radioactive substance is described by the equation N(t) = N0 e^(-λt), where N(t) is the amount of substance remaining at time t, N0 is the initial amount, and λ is the decay constant. Solving equations involving these exponential terms is essential for determining half-lives and predicting the behavior of radioactive materials.

2. Chemistry: Chemical kinetics, the study of reaction rates, often involves exponential functions. The rate of a first-order reaction, for example, follows the equation A(t) = A0 e^(-kt), where A(t) is the concentration of a reactant at time t, A0 is the initial concentration, and k is the rate constant. Solving equations involving these functions helps in determining reaction rates and understanding chemical processes.

3. Biology: Exponential functions model population growth and decay in biological systems. The growth of a bacterial population, for example, can be described by an exponential function. Similarly, pharmacokinetics, the study of how drugs are absorbed, distributed, metabolized, and excreted, often involves exponential decay curves. Solving equations involving these functions is crucial for understanding biological phenomena and developing medical treatments.

4. Engineering: In electrical engineering, exponential functions describe the behavior of circuits containing capacitors and inductors. The voltage across a capacitor during charging or discharging follows an exponential curve. Similarly, in mechanical engineering, damped oscillations and vibrations can be modeled using exponential functions. Solving equations involving these functions is essential for designing and analyzing engineering systems.

5. Finance: Exponential functions are fundamental in finance for calculating compound interest and modeling investments. The future value of an investment with continuous compounding is given by A = P e^(rt), where A is the future value, P is the principal, r is the interest rate, and t is the time. Solving equations involving these functions is critical for financial planning and investment analysis.

6. Environmental Science: Exponential functions are used to model the spread of pollutants in the environment and the decay of contaminants over time. Understanding these models helps in developing strategies for environmental protection and remediation.

The ability to solve equations involving exponential functions and radicals is therefore not just an academic exercise but a practical skill with wide-ranging applications. Mastering these techniques provides a valuable toolset for tackling real-world problems in various fields.

Conclusion

In this comprehensive exploration, we successfully solved the equation ${\sqrt{2e^x - 5} = 5 - \sqrt{2e^x}}$. We began by understanding the equation and identifying the domain restrictions imposed by the square roots. We then proceeded with a step-by-step solution, which involved squaring both sides, isolating the square root term, and solving for x. Our solution was found to be x = ln(9/2). We rigorously verified this solution by substituting it back into the original equation and checking it against the domain restrictions, confirming its validity.

We also explored an alternative approach using substitution, which provided a different perspective on the problem and simplified the algebraic manipulations. This method highlighted the flexibility in mathematical problem-solving and demonstrated how different techniques can lead to the same correct answer. Understanding multiple approaches can often provide deeper insights into the problem and make the solution process more efficient.

Furthermore, we discussed common mistakes that can occur when solving such equations, such as forgetting to check for extraneous solutions, ignoring domain restrictions, and making errors in algebraic manipulation. By identifying these potential pitfalls and providing strategies to avoid them, we aimed to enhance the reader's problem-solving skills and accuracy.

Finally, we underscored the real-world applications of solving exponential and radical equations across various fields, including physics, chemistry, biology, engineering, finance, and environmental science. This discussion emphasized the practical significance of the mathematical concepts and techniques explored, highlighting their relevance to real-world problems and applications.

By mastering the methods and strategies discussed in this article, readers can confidently tackle similar equations and apply these skills in diverse contexts. The combination of a systematic approach, attention to detail, and a thorough understanding of the underlying concepts are key to success in solving mathematical problems and in applying these solutions to real-world challenges. This exploration not only enhances mathematical proficiency but also fosters a deeper appreciation for the power and versatility of mathematical tools in addressing practical issues.