Discriminant in Quadratic Equations - Formula & Examples
Discriminant in Quadratic Equations - Formula & Examples
TL;DR
The discriminant Δ = b² - 4ac is the piece under the square root in the quadratic formula. Its sign tells you the nature of a quadratic's roots before you finish solving — two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or two complex roots (Δ < 0).
What Is the Discriminant in Math?
For a quadratic equation ax² + bx + c = 0 (with a ≠ 0), the discriminant is the expression:
Δ = b² - 4ac
It appears as the piece under the radical in the quadratic formula:
x = -b ± sqrt(b² - 4ac) / 2a = -b ± sqrt(Δ) / 2a
The discriminant's sign determines how many real solutions the equation has — without you doing any actual solving.
The Discriminant Formula
For ax² + bx + c = 0:
Δ = b² - 4ac
It's a single number you compute from the three coefficients. Always check that the equation is in standard form (ax² + bx + c = 0) before reading off a, b, c.
Worked example. Find the discriminant of 2x² - 5x + 1 = 0.
Here a = 2, b = -5, c = 1:
Δ = (-5)² - 4(2)(1) = 25 - 8 = 17
Since Δ > 0, the equation has two distinct real roots — no need to solve to know this.
What Are the Three Cases?
The sign of Δ determines the nature of the roots:
| Discriminant | Number of real roots | Geometric picture |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis at two points |
| Δ = 0 | One repeated real root | Parabola just touches x-axis at vertex |
| Δ < 0 | No real roots (two complex) | Parabola doesn't touch x-axis |
Case 1: Δ > 0 — Two Distinct Real Roots
Because Δ is a positive real number.
Case 2: Δ = 0 — One Repeated Root
The formula collapses to x = -b / 2a — a single value.
Case 3: Δ < 0 — No Real Roots
The square root of a negative number yields complex roots.
Three Worked Examples
Quick
For x² + 4x + 4 = 0, find the discriminant:
Δ = 16 - 16 = 0. One repeated real root at x = -2.
Standard
For what value of k does x² + kx + 9 = 0 have one repeated root?
k² - 36 = 0, giving k = ±6.
Stretch
For what values of k does kx² + 3x + 2 = 0 have two distinct real roots?
k < 9/8 and k ≠ 0.
Why Does the Discriminant Matter?
The discriminant tells you whether a real-world equation has real solutions:
- Physics — projectile motion.
- Engineering — stability analysis.
- Economics — equilibrium.
- Computer graphics — ray tracing.
A Worked Example
For what value of k does x² + (k-2)x + 4 = 0 have real and equal roots?
Correct method gives k = 6 or k = -2, both valid.
What Are the Most Common Mistakes With the Discriminant?
Mistake 1: Sign errors when b is negative
Ensure b² is always computed as a positive number.
Mistake 2: Forgetting to put the equation in standard form first
The standard form is necessary for correct calculation of Δ.
Mistake 3: Confusing "no real roots" with "no roots"
Negative discriminant means no real roots; complex solutions exist.
Key Takeaways
- The discriminant Δ = b² - 4ac gives vital information about the roots of quadratic equations.
- Importance of calculating Δ from equations in standard form.
A Practical Next Step
- Find the discriminant of 3x² + 4x + 2 = 0. How many real roots?
- For what value of k does x² + kx + 25 = 0 have one repeated root?
- For what values of k does x² + 2x + k = 0 have no real roots?
If problem 3 felt tricky, Δ < 0 means k > 1.