Mathematics — first biennium · Lesson
The discriminant Δ=b²−4ac predicts the number of real solutions and therefore the number of x-axis intersections of the parabola.
Prerequisites: Incomplete quadratics and radicals
Goal: Use the discriminant to predict the number of solutions and connect the quadratic formula to parabola zeros.
In this lesson: Use the discriminant to predict the number of solutions and connect the quadratic formula to parabola zeros.
New concept: The discriminant Δ=b²−4ac predicts how many real solutions ax²+bx+c=0 has: Δ>0 two, Δ=0 one double root, Δ<0 no real roots. The quadratic formula uses the same Δ under the square root.
How to proceed: Δ>0 means two intersections, Δ=0 one tangent point, and Δ<0 no x-axis intersection.
You can already assume: Incomplete quadratics and radicals.
Vary a, b and c to observe changes in concavity, vertex position and axis intersections.
With a = 0 the curve loses this family's characteristic shape; also try a non-zero value.
The parabola is tangent to the x-axis: Δ=0 and the real solution is double.
Completed exercises: 0/5 · Correct on first attempt: 0
Δ>0 means two intersections, Δ=0 one tangent point, and Δ<0 no x-axis intersection.
Data / situation: x²+4x+4=0
ConceptStep separates what is required from what is recommended or useful as reinforcement. Only required prerequisites block your progress.
Relationships: solid line = required · dashed line = recommended · dotted line = reinforcement.