Solving Exponential Equations: x^{x^2} x^{2x1}
In this article, we will explore the process of solving an exponential equation, specifically the equation (x^{x^2} x^{2x1}), and derive its solutions using algebraic methods and quadratic equations. This article is ideal for students and educators looking to strengthen their understanding of exponential and algebraic concepts. Let's dive into the steps and solutions in a detailed manner.
Step-by-Step Solution
Step 1: Initial EquationThe given exponential equation is (x^{x^2} x^{2x1}).
Step 2: SimplificationWe first simplify the equation by equating the exponents since the bases are the same ((x)). This leads to:
(x^2 2x 1)
Step 3: Forming a Quadratic EquationBy rearranging the terms, we form the standard quadratic equation:
(x^2 - 2x - 1 0)
Step 4: Solving the Quadratic EquationWe use the quadratic formula to solve the equation (x^2 - 2x - 1 0). The quadratic formula is given by:
(x frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a 1), (b -2), and (c -1).
Substituting these values into the formula, we get:
(x frac{2 pm sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} frac{2 pm sqrt{4 4}}{2} frac{2 pm sqrt{8}}{2} frac{2 pm 2sqrt{2}}{2} 1 pm sqrt{2})
Thus, the roots of the equation are (x 1 sqrt{2}) and (x 1 - sqrt{2}).
Step 5: Validating SolutionsWe need to verify if both solutions are valid by considering the original equation:
For (x 1 sqrt{2}) Both sides of the equation are well-defined and valid. For (x 1 - sqrt{2}) Both sides of the equation are also well-defined and valid.Summary of Solutions
The solutions to the equation (x^{x^2} x^{2x1}) are:
Valid Solutions: (x 1 sqrt{2}) (x 1 - sqrt{2})Conclusion
In this article, we solved the exponential equation (x^{x^2} x^{2x1}) using algebraic methods and quadratic equations. The solutions, as determined, are (x 1 sqrt{2}) and (x 1 - sqrt{2}). Understanding and solving such equations is crucial for advanced mathematical studies and applications. Stay tuned for more such articles and problems to enhance your mathematical skills.