b^2 - 4ac is more commonly known as the discriminant of the quadratic. The quadratic must first be in the form ax^2 + bx + c = 0 to be able to calculate this value, so let's rearrange:
2x^2 + 5x = 1
2x^2 + 5x - 1 = 0
Now we see that a = 2, b = 5 and c = -1. These can be put into the formula for the discriminant:
b^2 - 4ac = (5^2) - 4(2)(-1) = 25 + 8 = 33
Now, let's see what this actually means. The discriminant is useful for two main reasons. Firstly, it tells you how many real roots a quadratic has (i.e. how many times the curve intersects the x-axis in the x-y plane). If the discriminant is greater than zero, there are two real roots. If it is equal to zero, there is one real repeated root. If it is smaller than zero, there are two complex roots and no real ones.
We can see why this is the case if we look at the quadratic formula. I could derive it for you but it would be a bit superfluous, so for a quadratic in the standard form:
x = (-b +- sqrt(b^2 - 4ac))/2a
The key part is the plus-minus symbol. If the discriminant (the value inside the square root) has a positive nonzero value, the values of x when the symbol is plus and when it is minus will be different. If the discriminant is equal to zero, plus zero is the same as minus zero so we get one root. If the discriminant is negative, we cannot find a real solution to the square root so there are no real roots.
Another use is to work out whether the quadratic can be factorised rationally. This is achieved when the discriminant is a perfect square. Consider, if we square root a perfect square we get an integer value. This means the roots will be rational, and factorisation is nice. In any other case we would get irrational roots which have surd form.
Anyway that was just for interest and background, your answer is 33. See if you can work out:
a) how many real roots the quadratic has
b) does it factorise rationally?
I hope this helps you :)
