Problem 75 - 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. $$e^{x^{2}-3 x}=e^{x-2}$$

Step-by-Step Solution

Answer

The solutions to the equation \(e^{x^{2}-3 x}=e^{x-2}\) are \(x = 2 + \sqrt{2}\) which is approximately x = 3.414 and \(x = 2 - \sqrt{2}\) which is approximately x = 0.586 when rounded to three decimal places.

Step 1: Simplify the exponential equation

Since the bases of the exponential terms on either side of the equation are the same (\( e \)), we can set the exponents equal to each other. This simplifies the equation to: \n\(x^{2}-3x = x-2\)

Step 2: Transform equation into a quadratic equation

To transform the equation into a standard quadratic equation, subtract x and add 2 to both sides of the equation. The equation then becomes: \n\(x^2 - 4x + 2 = 0\)

Step 3: Solve the quadratic equation

To solve the quadratic equation, use the quadratic formula \((-b ± \sqrt{b^2-4ac})/(2a)\). In this equation, \(a = 1\), \(b = -4\), and \(c = 2\). Substituting these values into the formula gives: \n\[x = \frac{-(-4) ± \sqrt{(-4)^2-4*1*2}}{2*1}\] \n\[x = \frac{4 ± \sqrt{16-8}}{2}\] \n\[x = \frac{4 ± \sqrt{8}}{2}\] \nThe two solutions to the equation are \[x = 2 + \sqrt{2}\] and \[x = 2 - \sqrt{2}\] providing the decimal approximation gives solutions: x = 3.414 and x = 0.586.

Step 4: Verify with graphing utility

Lastly, you can use a graphing utility to confirm these solutions. By graphing the functions \(y=e^{x^{2}-3 x}\) and \(y=e^{x-2}\), it's observed that they intersect at x = 3.414 and x = 0.586, verifying the solutions found algebraically for x.

Key Concepts

Exponential Equation

An exponential equation is one in which variables appear in the exponent and can be solved by utilizing a variety of algebraic methods. As seen in the given exercise, e^{x^2-3x} = e^{x-2}, the process begins with identifying a common base for the exponential expressions — in this case, the natural exponential base, e.

When the bases are the same, you're able to set the exponents equal to each other and solve for the variable without worrying about the exponential part anymore. This is a fundamental technique when tackling exponential equations, as removing the exponent allows the problem to be simplified into a more familiar algebraic form, often leading to a quadratic equation, which is easier to solve.

Quadratic Formula

When faced with a quadratic equation, such as x^2 - 4x + 2 = 0, the quadratic formula is an invaluable tool for finding the roots. The formula, x = (-b ± √(b^2-4ac))/(2a), provides a standardized method for solving any quadratic equation. Here, a, b, and c correspond to the coefficients in the standard form of a quadratic equation, ax^2 + bx + c = 0.

The formula incorporates the discrimination, b^2-4ac, which indicates the nature of the roots. A positive discriminant suggests two real and distinct solutions, zero indicates a single repeated real solution, and a negative one indicates complex or imaginary solutions. After determining the values of a, b, and c, these are substituted into the formula to find the possible values of x.

Graphing Utility Verification

Verification of solutions through a graphing utility is an effective way to cross-check your work, providing a visual affirmation of the algebraically obtained results. After finding potential solutions to the equation, plotting the corresponding functions -- in our case, y=e^{x^2-3x} and y=e^{x-2} -- on a graphing calculator or computer software, allows us to observe the points of intersection.

These intersections represent the x-values for which the original exponential equation holds true. If the graphical intersections correspond to the solutions obtained algebraically, this serves as confirmation that the solutions are correct. For the technology-savvy student, this process is not just about confirmation but also provides a deeper understanding of the behavior and relationship of the functions involved.

Quadratic Equation Transformation

Transforming an equation into a quadratic form is often required to simplify the problem and make it solvable by the quadratic formula. In our exercise, the initial exponential equation is converted by setting the exponents equal and moving all terms to one side, thereby revealing the underlying quadratic equation, x^2 - 4x + 2 = 0.

This methodical transformation is critical as it changes an exponential problem, which might be complex and less intuitive, into a quadratic one, which is a format that students are usually more familiar with solving. Through such manipulations, problems are often made more manageable, paving the way for the application of well-established algebraic techniques like the quadratic formula, factorization, or completing the square.