Free maths tool · Level: Ages 14 to 16
Discriminant and parabola
Slide a, b and c and see what the discriminant really measures: whether the parabola cuts the x axis, touches it, or misses entirely.
The discriminant of ax² + bx + c is D = b² - 4ac. If D is positive the parabola cuts the x axis twice and there are two solutions. If D is zero it touches the axis and there is one. If D is negative the parabola stays wholly above or below and there are no real solutions.
The three cases are one motion
Your notes list three separate cases. On screen it is one continuous motion: raise c slowly and the parabola lifts, the two crossing points creep together, they merge, and then the parabola is clear of the axis.
The moment the two solutions merge is exactly the moment D hits zero. There is no separate rule for it, it is just the step in between.
Why the square root fixes the sign of D
The formula contains √D. A negative number has no real square root, and that is the entire reason a negative D means no solutions.
Set a to zero and the parabola becomes a straight line. The formula, with 2a on the bottom, stops working because you would divide by zero. Hence the condition a ≠ 0.
The vertex always sits between the roots
The axis of symmetry runs through the vertex at x = -b / (2a). When there are two solutions, the vertex is exactly halfway between them.
That is a quick check on any exercise: work out the roots, take their average, and you have the x of the vertex with no further work.
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Frequently asked questions
- How do you calculate the discriminant?
- With D = b² - 4ac, where a, b and c are the coefficients of ax² + bx + c = 0. Watch the signs on b and c, that is the most common slip.
- What does a discriminant of zero mean?
- The parabola touches the x axis at exactly one point. There is one solution and it coincides with the vertex. In the tool you can watch the two roots creep together into a single point.
- Why are there no solutions when D is negative?
- Because the formula needs the square root of D, and a negative number has no real square root. Graphically the parabola lies entirely above or entirely below the axis.
- What happens when a is zero?
- It stops being a quadratic and becomes a linear equation, and the graph becomes a straight line. Slide a to zero in the tool and the curvature disappears.