Quadratic Functions and Equations in One Variable
Nature of roots
Whether a quadratic has two, one or no real roots, decided by the discriminant b² − 4ac.
| English | Nature of roots |
|---|---|
| Bahasa Melayu | Jenis punca |
| 中文 | 根的性质 |
How it is used
For 2x² + 3x − 1 = 0, b² − 4ac = 3² − 4(2)(−1) = 9 + 8 = 17 > 0, so the nature of roots is two different real roots, and the parabola cuts the x-axis at two points.
Where it shows up in SPM
A standard exam skill (Form 4). Paper 1 asks you to state the nature of roots from the sign of b² − 4ac.
Paper 2 reverses it: given that an equation has equal roots or no real roots, you form an inequality in k and solve for the range of k.
Don't confuse it with
Open the chapter: Quadratic Functions and Equations in One Variable →
Frequently asked questions
What condition gives two equal roots?
Two equal roots occur exactly when b² − 4ac = 0. Geometrically the parabola touches the x-axis at one point instead of cutting it.
Many Paper 2 questions set b² − 4ac = 0 to find an unknown coefficient.
A question says the equation has no real roots, how do I set it up?
Write b² − 4ac < 0, substitute the coefficients (which usually contain k), and solve the resulting inequality for the range of k. For example x² + kx + 4 = 0 with no real roots gives k² − 16 < 0, so −4 < k < 4.