The discriminant
For ax² + bx + c = 0, the quantity b² − 4ac is called the discriminant. It is the part under the square root in the quadratic formula, so its sign decides the type of roots without solving the equation fully.
Three cases
- b² − 4ac > 0: two distinct real roots (the curve cuts the x-axis at two points).
- b² − 4ac = 0: two equal (repeated) real roots (the curve touches the x-axis).
- b² − 4ac < 0: no real roots (the curve does not meet the x-axis).
Many problems give a condition on the roots and ask for the range of an unknown constant; you translate the condition into an inequality or equation in the discriminant. The discriminant also links to the graph of the matching quadratic function: D > 0 means the parabola cuts the x-axis twice, D = 0 means it just touches (the x-axis is a tangent), and D < 0 means it stays entirely above or below the axis. This graphical reading is handy for questions phrased in terms of a line meeting or missing a curve.
Key formula
Discriminant D = b² − 4ac. Two distinct real roots ⇔ D > 0; equal roots ⇔ D = 0; no real roots ⇔ D < 0.
Worked example
Find the values of k for which x² + kx + 9 = 0 has two equal roots.
Equal roots ⇒ D = 0 ⇒ k² − 4(1)(9) = 0 ⇒ k² = 36 ⇒ k = ±6.
Check k = 6: x² + 6x + 9 = (x + 3)² = 0 gives the repeated root x = −3. ✓
Remember
- Identify a, b, c correctly (with their signs) before computing D.
- "Two distinct real roots" needs strictly D > 0; "real roots" allows D ≥ 0.
- D = 0 means one repeated value — the graph touches the x-axis.