Understanding the Positive Roots of the Equation (x^2 - 1^2 0)

In this article, we will explore the process of finding the roots of the equation (x^2 - 1^2 0), with a particular focus on identifying its positive roots. We'll go through a step-by-step algebraic method to understand the underlying concepts and ensure clarity in the solution.

Solving the Equation Algebraically

Step 1: Simplify the Equation Simplify the left side of the equation by treating it as a difference of squares. (x^2 - 1^2 0) Note that (1^2 1), so the equation simplifies to: (x^2 - 1 0) Step 2: Set Up the Equation for Solving To find the roots, set each factor equal to zero: (x^2 - 1 0) Or, equivalently: (x^2 - 1 0) Step 3: Factor the Equation The equation is now in a form that can be factored using the difference of squares formula: (x^2 - 1 (x - 1)(x 1) 0) This factorization comes from the identity (a^2 - b^2 (a - b)(a b)).

Identifying the Roots

Step 4: Solve Each Factor for x Set each factor equal to zero and solve for x: (x - 1 0) And: (x 1 0) Solving these equations gives: (x 1) or (x -1) Step 5: Determine the Positive Root The positive root of the equation is: (x 1)

Understanding the Quadratic Equation

It's important to recognize that the equation (x^2 - 1^2 0) is a quadratic equation, meaning it can be written in the form (ax^2 bx c 0), where (a 1), (b 0), and (c -1). Quadratic equations always have two roots, which can be found using the quadratic formula or by factoring, as demonstrated here.

Conclusion

In summary, the positive root of the equation (x^2 - 1^2 0) is (1). Understanding the algebraic process and applying the difference of squares formula are key steps in solving such equations. If you have any further questions or need additional help with quadratic equations or algebraic methods, feel free to reach out!