Quadratic Equations
12-4Graphs of Quadratic Equations: The Discriminant
Overview
Discriminant D = b² − 4ac tells how many real solutions (or x-intercepts) a quadratic has.
Key ideas
- D > 0: two real solutions.
- D = 0: one real solution (double root).
- D < 0: no real solutions.
Worked examples
Example 1
Problem. Discriminant of x² − 4x + 4.
Solution. D = 16 − 16 = 0; one solution.
Example 2
Problem. Discriminant of x² + x + 1.
Solution. D = 1 − 4 = −3; no real solutions.
Practice
Try each one. Click Show answer when ready.
- 1.
x² − 6x + 9: solutions?
- 2.
x² + 2x − 5: solutions?
- 3.
x² + x + 4: solutions?
Challenge problems
A little tougher — great for test prep. Click Show answer when ready.
- 1.
Find the discriminant of 3x² + 2x + 5 = 0 and determine the nature of the roots.
- 2.
For what values of k does x² + kx + 4 = 0 have exactly one real solution?