# 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

1. Find the discriminant of 3x² + 4x + 2 = 0. How many real roots?
2. For what value of k does x² + kx + 25 = 0 have one repeated root?
3. For what values of k does x² + 2x + k = 0 have no real roots?

If problem 3 felt tricky, Δ < 0 means k > 1.