*Given a function of the form #f(x) = ax^2 + bx + c#, one can find the zeroes of the function (that is, where #f(x) = 0#) by using the quadratic formula: #x = (-b +- sqrt(b^2-4ac))/(2a)#.
If the discriminant #b^2 - 4ac# is less than zero, these roots will be complex or imaginary, and if the discriminant is greater than or equal to 0. these roots will be real.
The use of #+-# informs us that there are two solutions here; one where we are subtracting #sqrt(b^2-4ac)#, and one where we are adding it.*
For a step by step example, assume the function #f(x) = x^2 - 7x + 10#. Here #a=1, b=-7, c=10.# If we set #f(x) = 0#, then the values of x for which #f(x) = 0#, that is to say the roots of #f(x)#, would be determined by the quadratic formula: #x = (-(-7)+- sqrt((-7)^2 -4(1)(10)))/(2(1)) = (7 +- sqrt(49-40))/2 = (7 +- sqrt 9)/2 = (7+-3)/2#. Thus, our roots will be at #x = (7+3)/2# and #x = (7-3)/2# or #x=5# and #x=2#.
On the graph of the function, we would see that the parabola crosses the x-axis at #x=2# and #x=5#.
