Final answer:
To solve the problem, apply the Pythagorean theorem with the given values, algebraically manipulate into a quadratic equation, and then solve for x using standard algebraic techniques such as factorization or the quadratic formula.
Step-by-step explanation:
To solve this problem, we use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is a² + b² = c². Given the values of a, b, and c as a = x, b = 2x - 1, and c = 2x + 1, we can substitute these into the Pythagorean theorem equation and solve for x.
By substitution, we get the equation x² + (2x - 1)² = (2x + 1)². After some algebraic manipulation, it becomes a quadratic equation in the form of ax² + bx + c = 0. We can then solve for x using the quadratic formula if necessary. The key is to expand the squares, collect like terms, and then solve for x through factorization or by using the quadratic formula. Remember to check the solutions in the original context of the problem, as sometimes extraneous solutions can arise when working with squares and square roots.