One, Two, or No Real Solutions

Property A quadratic graph reveals the number of real solutions: Two Real Solutions: The parabola intersects the x axis at two different points. One Real Solution: The parabola's vertex touches the x axis at a single point. No Real Solution: The parabola does not intersect the x axis at all.

Explanation Imagine a dolphin jumping out of the water! If it enters and exits the water, it hits the surface (the x axis) twice, giving two solutions. If it just taps the surface with its tail, that's one solution. If it jumps but never touches the surface at all, there is no real solution. The graph tells the whole story!

Examples The graph of $f(x) = x^2 9$ crosses the x axis twice, giving two real solutions: $x=3$ and $x= 3$. The graph of $f(x) = (x 5)^2$ only touches the x axis at its vertex, $(5, 0)$, so it has one real solution: $x=5$. The graph of $f(x) = x^2 + 2$ is entirely above the x axis and never crosses it, so it has no real solutions.