Question

What does it mean if a quadratic equation has no solution?

Answer 1

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 $\sqrt{b^2-4ac}$, where a,b and c are the coefficient of the equation $ax^2+bx+c=0$.

Since we can't compute the square roots of negative numbers using real numbers, if $b^2-4ac$ 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.