2.3 Discriminant and Roots

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Know the conditions for f(x) = 0 to have:
    • (i) Two real roots
    • (ii) Two equal roots
    • (iii) No real roots
  • Know the related conditions for a given line to:
    • (i) Intersect a given curve at two points
    • (ii) Be a tangent to a given curve
    • (iii) Not intersect a given curve at all
  • Understand how the discriminant connects to all of the above

📖 Main Content

https://cambridgeassistant.com

Part 1: What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are numbers, and a ≠ 0.

The solutions of a quadratic equation — the values of x that make the equation true — are called the roots of the equation.

To find the roots, you can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The ± symbol means you use + once and − once, which is why quadratics can have up to two roots.

Part 2: What is the Discriminant?

Look carefully at the quadratic formula. There is an expression inside the square root sign:

b2−4acb^2 - 4ac

This is called the discriminant. The word "discriminant" comes from "discriminate" — meaning it tells apart or distinguishes the different types of roots.

💡 The discriminant is your shortcut. Instead of solving the whole equation, you just calculate b2−4acb^2 - 4ac and check whether it is positive, zero, or negative. That tells you everything about the roots without solving for x.

Part 3: The Three Cases of the Discriminant

Case 1: b2−4ac>0b^2 - 4ac > 0 — Two Real Distinct Roots

When the discriminant is greater than zero (positive):

  • The square root exists and gives a positive number
  • Using + gives one root, using − gives a different root
  • So there are two different real roots
  • On a graph, the parabola crosses the x-axis at two separate points

What does "real" mean? A real number is any normal number you can place on a number line (like 3, −5, 0.7, etc.). It is not imaginary.

What does "distinct" mean? It simply means the two roots are different from each other.

Sign in to view full notes