When Parabolas Don't Touch

Algebra 2 · Axiom Academy

Discover why some quadratic equations have solutions you can't see on the graph — and what that means. A quadratic always has answers — even when the graph shows none Every parabola is the picture of a quadratic equation, and where it crosses the x-axis, those crossings are the solutions. But some parabolas float clear of the axis and never cross it. So do those equations simply have no answer? Watch first, then drive it yourself. Watch a parabola lift straight up. It starts below the axis, crossing at two points. As it rises those two crossings slide together, merge into one as it just kisses the axis, then the curve floats free and crosses nowhere — while the discriminant reading falls from positive, through zero, to negative. The two intercepts merging into one and then vanishing is the whole discriminant story in one motion. Drag the slider to shift the parabola up and down. When it crosses the x-axis, those crossing points are the solutions — so watch the graph, the discriminant, and the roots all move together. Here a=1 and b=0 , so b^2-4ac = -4c — the sign of the shift c alone decides how many real roots you get. Cross twice (shift below 0): the discriminant is positive , and the two visible x-intercepts are your two real roots . Just touch (shift exactly 0): the discriminant is zero . The two roots collapse into a single repeated root at the vertex — one solution that counts twice.

This is the written version of the interactive lesson above. See the full Algebra 2 course.