91Ó°ÊÓ

When dealing with exponential equations, it's often helpful to identify factors. In the context of the equation \( e^{x}(x+e)=0 \), factors are individual expressions multiplied together to form the entire equation.

• In this case, there are two factors: \( e^x \) and \( (x + e) \).
• An important property to note is that the product of two (or more) factors is zero only if at least one of the factors is zero.

This gives us a strategy to solve the equation, as we can analyze each factor individually to see if it leads to a solution. The factor \( e^x \) represents an exponential term. One of the intrinsic properties of exponential functions, specifically \( e^x \), is that they never reach zero. No matter what value \( x \) takes, \( e^x \) remains positive. This property indicates that the solution, if it exists, cannot come from this factor.The other factor, \( (x + e) \), is a linear expression. Linear expressions can easily be zero, which makes it a promising avenue for finding potential solutions by setting \( x + e = 0 \).

Finding the solutions for exponential equations involves setting individual factors to zero. Since we have the equation \( e^{x}(x+e)=0 \), we already established that \( e^x \) cannot be zero.
This means solutions must come from \( (x + e) = 0 \). Solving this linear equation involves simple algebraic manipulation:

• Set \( x + e = 0 \).
• To isolate \( x \), subtract \( e \) from both sides, leading to \( x = -e \).

This is the potential solution to the original problem. Note that in certain cases, equations may have multiple factors leading to multiple solutions, but in this specific problem, the only factor that can be zero is \( (x + e) \), resulting in \( x = -e \) being the sole solution.

Verification serves as a crucial step in solving equations, ensuring that solutions satisfy the original equation. After solving for \( x = -e \), it's important to plug this value back into the original equation \( e^{x}(x+e)=0 \) to verify its correctness.This process involves:

• Substituting \( x = -e \) into the equation.
• Evaluate the first term: \( e^{-e} \) remains a positive number, as exponential functions are never zero.
• Calculate \( (-e + e) = 0 \), so the entire product becomes \( e^{-e} \times 0 = 0 \).

The outcome, \( 0 = 0 \), verifies that our solution satisfies the equation since the left-hand side equals the right-hand side. Verification confirms that \( x = -e \) is indeed a correct solution, affirming both our algebraic manipulation and understanding of the function properties.