The number under the root
In the quadratic formula x = [−b ± √(b² − 4ac)]/(2a), the expression
b² − 4ac
is called the discriminant. It "discriminates" between the possible types of roots, without solving the equation.
What it tells you
| Discriminant | Number of real roots | Graph of y = ax² + bx + c |
|---|---|---|
| positive (> 0) | 2 distinct | crosses the x-axis twice |
| zero (= 0) | 1 (a repeated root) | touches the x-axis at the vertex |
| negative (< 0) | 0 | doesn’t touch the x-axis |
More detail when positive
- If the discriminant is a perfect square (like 1, 4, 9, 25), the roots are rational and the quadratic factors over the integers.
- Otherwise, the roots are irrational (they involve a square root).
Describing all of this is called finding the nature of the roots.
Examples
| Equation | b² − 4ac | Nature of the roots |
|---|---|---|
| x² − 5x + 6 = 0 | 25 − 24 = 1 | 2 rational roots |
| x² + 2x − 4 = 0 | 4 + 16 = 20 | 2 irrational roots |
| 4x² − 12x + 9 = 0 | 144 − 144 = 0 | 1 repeated root |
| x² + x + 3 = 0 | 1 − 12 = −11 | no real roots |
Counting x-intercepts
The number of x-intercepts of y = ax² + bx + c equals the number of real roots of ax² + bx + c = 0.
Finding unknown coefficients
Example: For what k does x² + kx + 9 = 0 have exactly one root?
- Discriminant = 0: k² − 36 = 0
- k = ±6
Example: For what values of k does 2x² + 4x + k = 0 have two real roots?
- 16 − 8k > 0 → k < 2
Example: When does kx² + 6x + 1 = 0 have no real roots?
- 36 − 4k < 0 → k > 9
Applications
Does a ball reach a certain height? h = −5t² + 20t + 1. Does it reach 25 m?
- −5t² + 20t + 1 = 25 → −5t² + 20t − 24 = 0
- Discriminant: 400 − 480 = −80 < 0 → No, the ball never reaches 25 m.
Does a line meet a parabola? Substitute and check the discriminant: 2 points, 1 point (tangent), or none.
Worked examples
Example 1: Nature of the roots of 3x² + 2x − 1 = 0? 4 + 12 = 16 → 2 rational roots.
Example 2: How many x-intercepts does y = x² − 4x + 5 have? 16 − 20 < 0 → none.
Example 3: Value of k for one root: x² − 8x + k = 0? 64 − 4k = 0 → k = 16.
Common mistakes
- Sign errors: with c negative, −4ac becomes positive.
- Taking the square root of the discriminant (you just need its sign).
- Forgetting to rearrange to standard form first.
Try it yourself
- Discriminant of 2x² − 3x + 5 = 0?
- Nature of the roots of x² − 10x + 25 = 0?
- How many x-intercepts does y = −x² + 3x + 4 have?
- Find k so that x² + 4x + k = 0 has two real roots.
Answers: 1) −31 (no real roots) 2) one repeated root (x = 5) 3) two 4) k < 4
Quick recap
- The discriminant b² − 4ac predicts the nature of the roots.
- Positive: 2 roots; zero: a repeated root; negative: no real roots.
- A perfect square discriminant means rational roots.