The Discriminant and the Nature of Roots

1. Review: The Quadratic FormulaLink to 1-review-the-quadratic-formula

Every quadratic equation can be written in standard form:

Its solutions are given by the quadratic formula:

The expression under the square root plays such an important role that it has its own name.

2. The DiscriminantLink to 2-the-discriminant

Because the discriminant appears under a square root, its sign determines everything about the roots.

3. Three Cases in DetailLink to 3-three-cases-in-detail

Case 1: $\Delta > 0$ — two real rootsLink to 3-three-cases-in-detail-case-1-delta-0-two-real-roots

The square root is a positive real number, so we get two distinct solutions:

Geometrically, the parabola crosses the $x$-axis at two points.

Case 2: $\Delta = 0$ — one repeated rootLink to 3-three-cases-in-detail-case-2-delta-0-one-repeated-root

Both solutions collapse into one:

The parabola just touches the $x$-axis at its vertex.

Case 3: $\Delta < 0$ — two complex rootsLink to 3-three-cases-in-detail-case-3-delta-0-two-complex-roots

The roots are complex conjugates:

The parabola never touches the $x$-axis.

4. ExamplesLink to 4-examples

5. Discriminant as a Graph DiagnosticLink to 5-discriminant-as-a-graph-diagnostic

The discriminant tells you whether a parabola intersects the $x$-axis without solving or sketching. Drag the slider to see how the graph and its roots change together.

6. A Parameter ProblemLink to 6-a-parameter-problem

Sometimes the roots must satisfy a condition, and the discriminant lets us solve for a parameter.

SummaryLink to summary

• The discriminant is $\Delta = b^2 - 4ac$.
• $\Delta > 0$: two real roots; $\Delta = 0$: one repeated root; $\Delta < 0$: two complex roots.
• The discriminant diagnoses how the parabola meets the $x$-axis.
• It is a powerful tool for parameter problems.