Problem 133 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$

Step-by-Step Solution

Answer

The solution to the equation \(-x e^{-x}+e^{-x}=0\) is \(x = 1\).

Step 1: Factor out the common terms

The exercise provides the equation \(-x e^{-x}+e^{-x}=0\). We can see the common term here, which is \(e^{-x}\), so it needs to be factored out. The equation becomes \(e^{-x}(1-x)=0\).

Step 2: Set each factor to zero

By the zero-product property, \(e^{-x} = 0\) or \(1 - x = 0\). However, \(e^{-x}\) can never be zero for any real number \(x\), so this equation has no solution.

Step 3: Solve for x

Solve the equation \(1 - x = 0\) for \(x\). Re-arranging terms gives us \(x = 1\).

Step 4: Verify using graphing utility

Using a graphing utility, we graph the functions \(y = e^{-x}(1-x)\) and \(y = 0\), we can see that they intersect at \(x = 1\), confirming that our solution is valid.

Key Concepts

Zero-Product Property

The zero-product property is a fundamental principle in algebra used to solve equations by setting each factor in a product equal to zero. It's particularly useful when dealing with polynomial or factored expressions. In mathematical terms, if a product of multiple factors is zero, at least one of the factors must itself be zero. This can be expressed as:

• If \(ab = 0\), then \(a = 0\) or \(b = 0\).

In applying this to the problem \(e^{-x}(1-x) = 0\), the property simplifies our task by allowing us to consider each part separately. However, because \(e^{-x}\) (the exponential function) is never zero for any real value of \(x\), it doesn’t contribute to solutions where the expression equals zero. Therefore, we turn our focus to \(1-x\), setting it to zero to find the solution \(x = 1\). This step harnesses the power of the zero-product property to quickly define possible solutions in algebraic equations.

Factorization

Factorization is the process of expressing an equation as a product of its simpler terms or factors. This method is essentially breaking down a complex expression into the smaller and more manageable components that, when multiplied together, form the original expression.In the original exercise, the equation \(-x e^{-x} + e^{-x} = 0\) was factorized by identifying the common term \(e^{-x}\), which appears in both terms of the equation. By factoring \(e^{-x}\), the equation simplifies to: \[e^{-x}(1-x) = 0\]Breaking an equation in this way is invaluable as it often reveals how to solve it by setting each factor equal to zero (as governed by the zero-product property). This simplification not only makes solving the equation more straightforward but also serves to visually clarify the structure of the algebraic expression.

Graphing Utility Verification

Graphing utility verification involves using a graphing calculator or software to visually confirm the solution of an equation. It's a handy method for double-checking your algebraic work and ensuring the validity of solutions obtained through calculation methods.To verify the solution \(x = 1\) of the equation \(e^{-x}(1-x) = 0\), a graph was created for the function \(y = e^{-x}(1-x)\) and the line \(y = 0\). Upon graphing, the intersection of the curve and the line at \(x = 1\) confirms the solution.Using graphing utilities allows us to:

• Observe solutions graphically.
• Verify the intersection points accurately.
• Develop a better understanding of the behavior of functions involved.

This step not only confirms algebraic solutions but also provides a visual representation, reinforcing the correctness of analytical work and enhancing conceptual understanding.