Final answer:
To solve the quadratic equation x² - 2x - 1 = 0, we use the quadratic formula. The solutions are found by identifying coefficients and performing arithmetic operations, resulting in x = 1 + √2 and x = 1 - √2.
Step-by-step explanation:
To solve the equation x² - 2x - 1 = 0 using the quadratic formula, we first identify the coefficients a = 1, b = -2, and c = -1. The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a).
We then substitute the values into the formula:
√(b² - 4ac) = √((-2)² - 4(1)(-1)) = √(4 + 4) = √8
x = (-(-2) ± √8) / (2(1))
x = (2 ± √8) / 2
x = 1 ± √2
Therefore, the solutions to the equation are x = 1 + √2 and x = 1 - √2.