The first graph we have here is a parabola which crosses the x-axis two times, which means there are two solutions to this equation, y = x^2 + x – 6, and those two solutions are the places where the parabola crosses the x-axis.

This one has two solutions, and those two solutions are 2 and -3. So, x = -3,2. Those are the two solutions.

The transcript is provided for your convenience

However, the parabola won’t always cross the x-axis two times. Sometimes, it may only intercept it one time, and when that happens, it’s the vertex that touches the x-axis one time. And so, this intersection, this parabola, this graph would only have one solution, and the solution here is the vertex where it intersects the x-axis at 3. So, here our solution would just be x = 3.

The third situation is when the parabola never touches the x-axis. That would be when you had two complex solutions, or no real solutions. And again, those would be two complex solutions, because it never actually touches the x-axis, so there aren’t any real solutions to the quadratic equation.

So, here are three different situations. You could have two solutions to your equation, one solution, or none at all, and you can tell what those solutions are just by looking at the graph of the parabola for the quadratic equation. You can see if it crosses twice, and those would be your two solutions. If it only intersects your x-axis one time, and that’s your one solution, or if it never touches the x-axis at all, and that means that the solutions are complex and they are not real.