Direct Answer
The discriminant is D = b² − 4ac from ax² + bx + c = 0. It reveals the root type before you solve: D > 0 → two distinct real roots; D = 0 → one repeated root; D < 0 → two complex roots. For x² − 5x + 6 = 0, D = 25 − 24 = 1 → two real roots (x = 2 and x = 3).
Where the Discriminant Comes From
In the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, only the part under the root decides the shape of the answer: b² − 4ac. Positive → two answers from ±; zero → branches collapse into one; negative → complex answers.
The Discriminant Decision Table
| D = b² − 4ac | Roots | Graph vs x-axis | Example |
|---|---|---|---|
| D > 0 | Two distinct real roots | Crosses twice | x² − 5x + 6 = 0 (D = 1 → x = 2, 3) |
| D = 0 | One repeated root | Touches once (vertex) | x² + 2x + 1 = 0 (D = 0 → x = −1) |
| D < 0 | Two complex roots | Never touches | x² + x + 5 = 0 (D = −19) |
Worked Example 1: D > 0 — Two Real Roots
x² − 5x + 6 = 0. D = (−5)² − 4(1)(6) = 25 − 24 = 1 → two real roots (factors: (x − 2)(x − 3) → x = 2, 3; vertex at x = 2.5).
Worked Example 2: D = 0 — One Repeated Root
x² + 2x + 1 = 0. D = 4 − 4 = 0 → x = −b/2a = −1. Parabola touches the axis once at (−1, 0).
Worked Example 3: D < 0 — Two Complex Roots
x² + x + 5 = 0. D = 1 − 20 = −19 → roots (−1 ± i√19)/2 ≈ −0.5 ± 2.18i. Not “no solution” — no real solution.
Why Calculate the Discriminant First?
- Predicts factoring success: perfect-square D → factors over rationals — solving by factoring will work.
- Catches coefficient errors: a five-second sanity check.
- Answers “how many solutions?” directly: many questions need only the count. Calcrio’s free quadratic equation solver computes D automatically.
Reading the Discriminant Off a Graph
Crosses twice → D > 0; touches once → D = 0; floats without touching → D < 0.
Quick-Fire Practice
- x² − 4x + 4 = 0 → D = 0 → one repeated root (x = 2).
- x² − 3x − 4 = 0 → D = 25 → two real roots (x = 4, x = −1; factors).
- x² + 6x + 10 = 0 → D = −4 → two complex roots (−3 ± i).
- 3x² − 5x + 2 = 0 → D = 1 → two real roots (x = 1, x = 2/3).
- x² − 8x + 16 = 0 → D = 0 → one repeated root (x = 4).
What Test Questions Actually Ask
- “How many real solutions does x² − 3x − 4 = 0 have?” → D = 25 > 0 → two (4 and −1).
- “For what k does x² + kx + 9 = 0 have exactly one solution?” → D = 0: k² − 36 = 0 → k = ±6.
- “Show x² + 6x + 10 = 0 has no real roots.” → D = −4 < 0 → done.
Common Discriminant Mistakes
Using b instead of b². D = 4 − 4 = 0, not 2 − 4 = −2. Dropping b’s sign: (−5)² = 25. Forgetting the 4. Treating D < 0 as "no solution".
FAQ
How do you find the discriminant? D = b² − 4ac. x² − 5x + 6 → D = 25 − 24 = 1.
What does D = 1 mean? Two distinct real roots; perfect square → factors.
Can the discriminant be a decimal? Yes — only the sign matters.
Why is it called the discriminant? It discriminates between the three root types.
Does it work if a = 0? No — that’s linear. (Calcrio’s quadratic formula solver flags a = 0.)
Related: How to Solve Quadratic Equations by Factoring (5 Steps)