Quadratic Discriminant
The discriminant b²−4ac determines the nature and number of roots of a quadratic equation.
Key to solving quadratics.
The Formula
The discriminant is the expression under the square root in the quadratic formula. For a quadratic equation ax² + bx + c = 0, the discriminant determines how many real solutions exist.
Variables
| Symbol | Meaning |
|---|---|
| D | The discriminant (also written as Δ in some textbooks) |
| a | Coefficient of x² (must not be zero) |
| b | Coefficient of x |
| c | Constant term |
Interpreting the Discriminant
| Value of D | Number of Real Roots | Nature of Roots |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses x-axis at two points |
| D = 0 | One repeated real root | Parabola touches x-axis at exactly one point (vertex on axis) |
| D < 0 | No real roots (two complex roots) | Parabola does not cross the x-axis |
If D is a perfect square and the coefficients are integers, the quadratic can be factored over the integers.
Connection to the Quadratic Formula
The discriminant appears inside the square root. When D is negative, the square root of a negative number gives complex (imaginary) solutions.
Example 1
How many real solutions does 2x² + 5x − 3 = 0 have?
Identify: a = 2, b = 5, c = −3
Calculate: D = 5² − 4(2)(−3) = 25 + 24
D = 49 > 0 — Two distinct real roots (and since 49 is a perfect square, the equation factors neatly)
Example 2
Analyze x² + 4x + 4 = 0
Identify: a = 1, b = 4, c = 4
Calculate: D = 4² − 4(1)(4) = 16 − 16
D = 0 — One repeated real root (x = −2, the parabola's vertex touches the x-axis)
Example 3
Does x² + 2x + 5 = 0 have real solutions?
Identify: a = 1, b = 2, c = 5
Calculate: D = 2² − 4(1)(5) = 4 − 20
D = −16 < 0 — No real roots. The solutions are complex: x = −1 ± 2i
When to Use It
Use the discriminant to quickly analyze quadratic equations without fully solving them.
- Determining how many real solutions a quadratic equation has
- Deciding whether a quadratic can be factored over the integers (D must be a perfect square)
- Analyzing whether a parabola intersects the x-axis (and how many times)
- Solving optimization problems that lead to quadratic equations
- Finding conditions on parameters that give a desired number of solutions