Question 154861
In general, the number of solutions for a polynomial is equal to the degree of the polynomial.
A quadratic equation is a polynomial of degree 2 so it would have 2 solutions. The type of solutions a quadratic equation can be determined by examining the discriminant: {{{(b^2-4ac)}}} which is taken from the quadratic formula:{{{x = -b+-sqrt(b^2-4ac))/2a}}} 
If the discriminant is negative, there are no real solutions/roots. This makes sense when you realize that a negative discriminant (the square root of a negative quantity) will yield complex solutions.
If the discriminant is zero, there is one real solution/root, sometimes referred to as a double root because you get two real solutions that are identical.
If the discriminant is positive, there are two real solutions/roots.
It is helpful to look at the graphs of quadratic equations with the above type of solutions/roots:

{{{graph(400,400,-5,5,-5,5,2x^2+x+3,x^2+6x+9,x^2-5x+2)}}}
Green graph: {{{y = 2x^2+x+3}}} Discriminant is negative, no real roots.
Red graph: {{{y = x^2+6x+9}}} Discriminant is zero, one double root.
Blue graph: {{{y = x^2-5x+2}}} Discriminant is positive, two real root.
As you can see, the roots or solutions to these equations are the x-values where the curves (parabolas) intersect the x-axis.