determine the nature of the solutions of the equation x^2 - 20 = 0

To determine the nature of the solutions of the equation x^2 - 20 = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation x^2 - 20 = 0, a = 1, b = 0, and c = -20. Plugging these values into the quadratic formula, we get:

x = (-0 ± √(0^2 - 4(1)(-20))) / (2*1)

Simplifying further, we have:

x = (± √(0 + 80)) / 2
x = (± √80) / 2
x = ± √(80) / 2
x = ± √(16*5) / 2
x = ± (±4√5) / 2
x = ± 2√5

From the above calculations, we can see that the equation x^2 - 20 = 0 has two real solutions: x = 2√5 and x = -2√5. These solutions are irrational since they involve the square root of 5. Therefore, the nature of the solutions of the given equation is two distinct real solutions.