How to determine the number of unique real roots of a quadratic equation.

Take any quadratic equation, eg/ 3x²+4x-2=5, and rearrange to equal 0, ie/ 3x²+4x-7=0   (if you have an expression, ie/ there is no equals sign, then simply equate the expression to 0).

Now, we use the discriminant function, b²-4ac, of the quadratic, ax²+bx+c=0. Notice that a=3, b=4, and c=-7, in this case. This means that the discriminant is 4²-43(-7)=16-(-84)=100. This is greater than 0. Therefore, there exist 2 unique real roots to our quadratic.

Simply put, if, for any quadratic of the form ax²+bx+c=0, that b²-4ac>0, then there exist 2 unique real roots, if b²-4ac=0 then there is 1 repeated real root, and if b²-4ac<0, then there are no real roots.