Discriminant and Nature of Roots

The Discriminant

In the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root is called the discriminant:

$D = b^2 - 4ac$

Why Does It Matter?

The discriminant tells you what kind of roots you’ll get, without solving.

The sign of D determines the nature of the roots.

Three Cases

Case 1: $D > 0$

Square root of a positive → Two distinct real roots

Case 2: $D = 0$

Square root of zero → One repeated real root

The quadratic is a perfect square.

Case 3: $D < 0$

Square root of a negative → No real roots

The roots are complex numbers.

Summary

Example 1: Two Distinct Roots

Equation: $x^2 + 5x + 6 = 0$

$D > 0$ → Two distinct real roots.

Solving gives $x = -2$ and $x = -3$.

Example 2: One Repeated Root

Equation: $x^2 - 6x + 9 = 0$

$D = 0$ → One repeated root.

This is a perfect square: $(x - 3)^2 = 0$, so $x = 3$.

Example 3: No Real Roots

Equation: $x^2 + x + 1 = 0$

$D < 0$ → No real roots.

Bonus: Rational vs Irrational Roots

When $D > 0$:

• If $D$ is a perfect square (1, 4, 9, 16, …) → roots are rational
• If $D$ is not a perfect square → roots are irrational

Example: $x^2 - 2x - 1 = 0$

$D = 4 + 4 = 8$

$D > 0$ but 8 is not a perfect square → roots are irrational ($1 \pm \sqrt{2}$).