From an analytical point of view, a quadratic equation has no (real) solution if the discriminant is negative.
In fact, the equation for the solution involves the quantity [tex] \sqrt{b^2-4ac} [/tex], where a,b and c are the coefficient of the equation [tex] ax^2+bx+c=0 [/tex].
Since we can't compute the square roots of negative numbers using real numbers, if [tex]b^2-4ac<0 [/tex] the equation has no solution.
From a geometric point of view, the solutions of an equation are the points where the graph of the equation intersects the x axis. So, if a parabola (i.e. the graph of a quadratic polynomial) has no solutions, it means that its graph never intersercts the x axis.
