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# The discriminant
- URL: https://www.esheets.io/the-discriminant/
- Published: 2025-06-04T20:19:28.000Z
- Updated: 2026-06-21T16:47:17.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

Ever wondered how you can tell if a quadratic equation will have two solutions, one, or none—without solving it? That’s where the discriminant comes in. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise using the discriminant with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Identifying a, b and c.
- Calculating b² − 4ac.
- Interpreting whether there are two, one or no real roots.
- Using the sign of the discriminant.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Topic guide

### What this worksheet practises

This worksheet focuses on calculating and interpreting the "discriminant" of a quadratic equation (ax² + bx + c = 0). The discriminant is a quick test that tells you exactly how many roots (solutions) the equation has, without having to actually solve the whole thing.

### Key method

The formula for the discriminant is a small part of the main Quadratic Formula: **b² − 4ac**.

- Identify the values for **a**, **b**, and **c** from your quadratic equation. (Ensure the equation equals zero first).
- Substitute these values carefully into the formula **b² − 4ac**.
- Calculate the final number.
- **Interpret the result:**  
  - If the answer is **Positive** (greater than zero), the equation has **two distinct real roots** (it crosses the x-axis twice).
  - If the answer is exactly **Zero**, the equation has **one repeated real root** (it just touches the x-axis once).
  - If the answer is **Negative** (less than zero), the equation has **no real roots** (it floats above or below the x-axis and never crosses it).

### Worked example

**Use the discriminant to determine the number of real roots for the equation 3x² − 5x + 4 = 0.**

Step 1: Identify a, b, and c.

a = 3, b = −5, c = 4.

Step 2: Substitute into b² − 4ac. Always use brackets for negative numbers!

(−5)² − (4 × 3 × 4)

Step 3: Calculate.

25 − (48)

25 − 48 = −23.

Step 4: Interpret the result.

Because the discriminant is negative (−23), there are **no real roots**.

### Common mistakes to avoid

The single most common error is miscalculating the 'b²' part when 'b' is a negative number. If you type -5² into a calculator without brackets, it gives -25\. A squared number must always be positive. You must type (-5)² to get +25\. This mistake will completely change your final interpretation.

### Things to remember

The discriminant is just the bit that lives "inside the square root" of the full quadratic formula. The rules make logical sense: you cannot square root a negative number (no roots), the square root of zero is just zero (one root), and the square root of a positive number gives a ± result (two roots).