Using the Discriminant to Determine Number of Solutions
The discriminant Δ = b² − 4ac of a quadratic equation predicts the number of intersection points between a line and a parabola in a linear-quadratic system, as taught in Grade 11 enVision Algebra 1 (Chapter 9: Solving Quadratic Equations). After substitution or elimination produces a quadratic ax² + bx + c = 0, calculate the discriminant: Δ > 0 gives two intersection points, Δ = 0 means the line is tangent to the parabola (one point), and Δ < 0 means no intersection. This connects algebra to the geometry of the system.
Key Concepts
For a linear quadratic system, after substitution or elimination, you get a quadratic equation $ax^2 + bx + c = 0$. The discriminant $\Delta = b^2 4ac$ determines the number of solutions: If $\Delta 0$: 2 real solutions (line intersects parabola at 2 points) If $\Delta = 0$: 1 real solution (line is tangent to parabola) If $\Delta < 0$: 0 real solutions (line does not intersect parabola).
Common Questions
What does the discriminant tell you about a linear-quadratic system?
The discriminant b² − 4ac of the resulting quadratic equation tells you how many points the line and parabola share: 2 (Δ > 0), 1 (Δ = 0), or 0 (Δ < 0).
What does Δ > 0 mean geometrically?
The line intersects the parabola at two distinct points.
What does Δ = 0 mean geometrically?
The line is tangent to the parabola — it touches at exactly one point.
What does Δ < 0 mean geometrically?
The line does not intersect the parabola — there are no real solutions.
When does the discriminant apply in a linear-quadratic system?
After substituting the linear equation into the quadratic (or using elimination), you get a quadratic equation. Calculate the discriminant of that equation to predict the number of solutions.
Why is computing the discriminant useful before solving completely?
It tells you how many solutions to expect, helping verify your work and identify when the system has no real solutions before spending time calculating them.