Final answer:
To solve the equation 2x^2 - 5x - 1 = 0, we use the quadratic formula, where the coefficients are a=2, b=-5, and c=-1, to find two distinct real solutions.
Step-by-step explanation:
To solve the quadratic equation 2x^2 - 5x - 1 = 0 using the quadratic formula, we first identify the coefficients a, b, and c from the equation, where a = 2, b = -5, and c = -1. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). Substituting our values into this formula gives us:
x = (5 ± √((-5)^2 - 4(2)(-1))) / (2(2))
x = (5 ± √(25 + 8)) / 4
x = (5 ± √(33)) / 4
Thus, we have two possible solutions for x, which are:
x = (5 + √33) / 4
x = (5 - √33) / 4
Therefore, the given quadratic equation has two distinct real solutions.