How Many Times Does It Cross?
Drag the parabola's vertex anywhere — sideways moves the crossings without changing how many there are; up and down is what matters. It crosses the x-axis twice, then just touches it once, then not at all, and b² − 4ac changes sign at exactly that moment.
b² − 4ac = 16 − 4(1)(0) = 16
Positive, so the parabola crosses the x-axis twice — at x = 0 and x = 4.
- Equation
- x² − 4x + 0
- Discriminant
- 16
- Real solutions
- 2
Drag the marked vertex anywhere, or use the arrow keys.
Ready to try some? Discriminant worksheets
Where it usually goes wrong
b² − 4ac is memorized as a sorting rule — positive, zero or negative, two answers or one or none — with no picture attached, so a child who mis-substitutes a negative b and gets the sign wrong has no way to notice. It is not an arbitrary sorting rule: it is the height of the parabola's own lowest point above or below the x-axis, measured in the equation's own units, and a picture of that height is a picture of the sign.
Try this
- Drag the vertex down until the parabola crosses the axis twice. Read the discriminant — positive, matching two crossings.
- Drag it slowly up until the two crossings merge into one point sitting on the axis. The discriminant reads exactly zero at that instant, not a moment before or after.
- Keep dragging up past that point. The crossings vanish, the parabola sits entirely above the axis, and the discriminant is negative.
Where this goes next
Every quadratic formula problem starts by finding this same number, and knowing what it looks like is what makes 'no real solution' read as a fact about a picture rather than a sign someone got wrong.