The Quadratic Formula Explained

The quadratic formula is a fundamental tool in algebra used to solve quadratic equations, which are equations of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero.

The formula provides the values of 'x' that satisfy the equation. It is given by:

x = [-b ± √(b² - 4ac)] / 2a

Understanding the Discriminant

The term inside the square root, (b² - 4ac), is called the discriminant. It is often denoted by the Greek letter Delta (Δ).

The discriminant tells us about the nature of the roots (solutions) of the quadratic equation:

• If Δ > 0: There are two distinct real roots.
• If Δ = 0: There is exactly one real root (a repeated root).
• If Δ < 0: There are two complex conjugate roots (no real roots).

Example

Let's solve the quadratic equation: x² + 5x + 6 = 0

Here, a = 1, b = 5, and c = 6.

First, calculate the discriminant:

Δ = b² - 4ac = (5)² - 4(1)(6) = 25 - 24 = 1

Since Δ > 0, there are two distinct real roots.

Now, apply the quadratic formula:

x = [-5 ± √(1)] / 2(1)

x = [-5 ± 1] / 2

This gives us two solutions:

• x₁ = (-5 + 1) / 2 = -4 / 2 = -2
• x₂ = (-5 - 1) / 2 = -6 / 2 = -3

So, the solutions to the equation x² + 5x + 6 = 0 are x = -2 and x = -3.