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Exponential functions are mathematical expressions in which a variable appears as an exponent. They take the form of \( a^x \), where 'a' is a constant base and 'x' is the exponent. These functions grow rapidly and are widely used in various fields such as finance, biology, and physics.
In the given exercise, we have a specific type of exponential function: \( e^{x^2 - x} \). Here, 'e' is the base, which is a mathematical constant approximately equal to 2.71828. The exponent is \( x^2 - x \). To solve equations involving exponential functions, we often look for ways to simplify the exponent.
In our example, we simplify the expression \( e^{x^2 - x} = 1 \) by recognizing that \( e^0 = 1 \). By setting the exponent equal to zero, we transform this into a simpler quadratic problem.

Quadratic equations are polynomial equations of the second degree and have the general form \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants. Solving for 'x' requires finding the values that satisfy the equation.
In our exercise, once we set the exponent \( x^2 - x \) equal to zero, we end up with a quadratic equation: \( x^2 - x = 0 \). This is a simpler form of a quadratic equation where 'c' is zero.
To solve a quadratic equation, we can use various methods such as factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward approach.

Factoring is a technique used to simplify polynomial expressions by expressing them as a product of simpler polynomials. It's especially useful in solving quadratic equations.
For the quadratic equation \( x^2 - x = 0 \), factoring allows us to write this as \( x(x - 1) = 0 \). This step breaks the equation into two simpler parts. Each factor can then be set to zero: \( x = 0 \) or \( x - 1 = 0 \).
Solving these two linear equations gives us the solutions \( x = 0 \) and \( x = 1 \). This demonstrates that factoring is a powerful tool in solving quadratics, making the process more manageable.

• Start by looking for common factors.
• Write the equation as a product of simpler expressions.
• Set each factor to zero to solve for the variable.