Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{-x^{2}}=e^{x^{2}-2 x}$$

Step-by-Step Solution

Answer

The solution to the exponential equation is \(x = 0, 1\).

Step 1: Equate the exponents

Both sides of the equation have the same base ('e'), so we can say that the exponents must be equal, i.e. \(-x^{2} = x^{2}-2x\).

Step 2: Simplify to a Quadratic equation

Rearrange the equation so that all terms are on one side; this results in a quadratic equation: \(2x^{2} - 2x = 0\).

Step 3: Solve the quadratic equation

Factor out the equation to find roots of x : \(2x (x - 1) = 0\). Set each factor to zero and solve, to get \(x = 0, 1\).

Step 4: Verify your answers

Using a graphing utility, graph the functions \(y = e^{-x^{2}}\) and \(y = e^{x^{2}-2x}\) and see that they intersect at x = 0 and x = 1. This verifies that the roots found algebraically are correct.

Key Concepts

Understanding Exponential Equations

Exponential equations are mathematical expressions where the unknown variable appears in the exponent. A classic example, and one that appears in many textbooks, is the equation involving Euler's number, which is known as the natural base, 'e'. An equation such as \(e^{-x^{2}} = e^{x^{2} - 2x}\) has its variable 'x' in the exponent on both sides of the equation.

When solving such equations, we use the property which states if \(a^{x} = a^{y}\) , then \(x = y\b\) (given \(aeq0\b\) and \(aeq1\b\)). This property simplifies the process by reducing the exponential equation to a more familiar algebraic form. The key to solving these equations is to express both sides of the equation with the same base and then equate the exponents, leading to a more straightforward equation.

Exercise Improvement Advice

One effective way to grasp this concept is to practice transforming various exponential equations into their algebraic counterparts and then solving them. Additionally, understanding logarithms can further benefit students in solving more complex exponential equations where equating bases is not as straightforward.

Solving Quadratic Equations

Quadratic equations follow a standard form: \(ax^2 + bx + c = 0\b\) where 'a', 'b', and 'c' are known coefficients, and 'x' represents the unknown variable. The process of solving the quadratic equation typically includes factoring, using the quadratic formula, completing the square, or employing a graphing utility to visualize the roots.

Factorisation is the method used in the provided step-by-step solution, where the equation \(2x^2 - 2x = 0\b\) is factored as \(2x(x - 1)= 0\b\) to reveal the roots: x = 0 and x = 1.

Understanding how to solve quadratic equations is critical in algebra as these equations appear frequently in various mathematical contexts. Mastery in this area involves being able to handle diverse forms of quadratic equations and employing the most efficient method to solve them.

Exercise Improvement Advice

To improve, students should focus on recognizing patterns that allow for quick factoring and also practice employing the quadratic formula when factorisation is not possible. The quadratic formula provides a reliable way to find roots and can also be used to verify factored results.

Graphing Utility Verification

Verification of algebraic solutions with a graphing utility adds an extra layer of confirmation that the solutions are correct. It involves plotting the equations on a graph to visually ascertain the points at which they intersect, which corresponds to the solutions of the equation.

In the given exercise, the student is instructed to plot \(y = e^{-x^{2}}\b\) and \(y = e^{x^{2} - 2x}\b\) to confirm the algebraically found roots. If done correctly, the graphs will intersect at x = 0 and x = 1, which verifies the roots. This visual approach often aids in comprehensive understanding and can reveal errors not evident through algebraic manipulation.

Exercise Improvement Advice

To benefit fully from graphing utilities, students should be encouraged to explore how small changes in an equation affect its graph. Understanding the relationship between algebraic manipulation and the resulting graphical changes can improve a student's overall mathematical intuition and problem-solving skills.