Solve For \[$ X \$\] In The Equation:$\[ X^2 E^x + X E^x - E^x = 0 \\]

Feb 28, 2025 by ADMIN 71 views

**Solving Exponential Equations: A Step-by-Step Guide to Finding the Value of x**

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve, but with the right approach, you can find the value of x that satisfies the equation. In this article, we will focus on solving the equation $x^2 e^x + x e^x - e^x = 0$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $x^2 e^x + x e^x - e^x = 0$ . This equation involves an exponential function $e^x$ , which is a fundamental concept in mathematics. The equation can be rewritten as $e^x (x^2 + x - 1) = 0$ . This form of the equation makes it easier to identify the possible solutions.

Factoring the Quadratic Expression

The quadratic expression $x^2 + x - 1$ can be factored as $(x + 1)(x - 1)$ . Therefore, the equation can be rewritten as $e^x (x + 1)(x - 1) = 0$ . This form of the equation makes it clear that the solutions are the values of x that make the expression $(x + 1)(x - 1)$ equal to zero.

Finding the Solutions

To find the solutions, we need to set each factor equal to zero and solve for x. The first factor is $e^x$ , which is always positive and never equal to zero. Therefore, we can ignore this factor and focus on the other two factors.

Solving for x

The first factor is $(x + 1) = 0$ , which gives us $x = -1$ . The second factor is $(x - 1) = 0$ , which gives us $x = 1$ . Therefore, the solutions to the equation are $x = -1$ and $x = 1$ .

Checking the Solutions

To verify that these solutions are correct, we can plug them back into the original equation. If the equation holds true for both values of x, then we can be confident that we have found the correct solutions.

Conclusion

In this article, we have solved the equation $x^2 e^x + x e^x - e^x = 0$ using a step-by-step approach. We have factored the quadratic expression, identified the possible solutions, and verified that the solutions are correct. The solutions to the equation are $x = -1$ and $x = 1$ . This example demonstrates the importance of breaking down complex equations into manageable steps and using algebraic techniques to find the solutions.

Applications of Exponential Equations

Exponential equations have numerous applications in various fields, including physics, engineering, and economics. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments. In physics, exponential equations are used to describe the behavior of particles in quantum mechanics and the expansion of the universe.

Tips for Solving Exponential Equations

When solving exponential equations, it is essential to follow these tips:

Identify the base: The base of the exponential function is a critical component of the equation. Make sure to identify the base and understand its properties.
Use algebraic techniques: Exponential equations can be challenging to solve, but algebraic techniques such as factoring and substitution can help simplify the equation.
Check the solutions: Once you have found the solutions, plug them back into the original equation to verify that they are correct.

Common Mistakes to Avoid

When solving exponential equations, it is essential to avoid these common mistakes:

Ignoring the base: The base of the exponential function is a critical component of the equation. Make sure to identify the base and understand its properties.
Not checking the solutions: Once you have found the solutions, plug them back into the original equation to verify that they are correct.
Using incorrect algebraic techniques: Exponential equations can be challenging to solve, but algebraic techniques such as factoring and substitution can help simplify the equation.

Conclusion

In conclusion, solving exponential equations requires a step-by-step approach and a clear understanding of algebraic techniques. By following the tips and avoiding common mistakes, you can find the value of x that satisfies the equation. Exponential equations have numerous applications in various fields, and understanding how to solve them is essential for success in mathematics and science.

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. Exponential functions are functions that involve the variable x as an exponent, such as $e^x$ or $2^x$ .

Q: How do I identify the base of an exponential equation?

A: The base of an exponential equation is the number or expression that is raised to the power of x. For example, in the equation $e^x$ , the base is $e$ . To identify the base, look for the number or expression that is being raised to the power of x.

Q: What is the difference between an exponential equation and a polynomial equation?

A: An exponential equation involves an exponential function, while a polynomial equation involves a polynomial expression. For example, the equation $x^2 + 3x - 4 = 0$ is a polynomial equation, while the equation $e^x + 2e^x - 3e^x = 0$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, follow these steps:

Identify the base: Identify the base of the exponential function.
Use algebraic techniques: Use algebraic techniques such as factoring and substitution to simplify the equation.
Check the solutions: Once you have found the solutions, plug them back into the original equation to verify that they are correct.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

Ignoring the base: Make sure to identify the base of the exponential function and understand its properties.
Not checking the solutions: Once you have found the solutions, plug them back into the original equation to verify that they are correct.
Using incorrect algebraic techniques: Use algebraic techniques such as factoring and substitution to simplify the equation.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

Population growth: Exponential equations are used to model population growth and decline.
Chemical reactions: Exponential equations are used to describe the behavior of particles in chemical reactions.
Financial investments: Exponential equations are used to model the growth of investments over time.

Q: How do I graph an exponential function?

A: To graph an exponential function, follow these steps:

Identify the base: Identify the base of the exponential function.
Determine the direction: Determine the direction of the graph, which is determined by the base.
Plot points: Plot points on the graph to visualize the function.

Q: What is the difference between an exponential function and a logarithmic function?

A: An exponential function involves an exponential expression, while a logarithmic function involves a logarithmic expression. For example, the function $e^x$ is an exponential function, while the function $\log(x)$ is a logarithmic function.

Conclusion

In conclusion, solving exponential equations requires a clear understanding of algebraic techniques and a step-by-step approach. By following the tips and avoiding common mistakes, you can find the value of x that satisfies the equation. Exponential equations have numerous real-world applications, and understanding how to solve them is essential for success in mathematics and science.