How to Solve Quadratic Equations Step by Step

1. The Standard Form of a Quadratic Equation

Any second-degree polynomial equation can be rearranged into the standard form:

ax² + bx + c = 0 (where a ≠ 0)

2. The Quadratic Formula

When an equation cannot be factored easily by inspection, the quadratic formula provides universal solutions for all values of a, b, and c:

x = (−b ± √(b² − 4ac)) / (2a)

3. Discriminant Analysis (Δ = b² − 4ac)

The expression beneath the radical sign is called the discriminant:

• Δ > 0: The equation has two distinct real roots (parabola crosses the X-axis twice).
• Δ = 0: The equation has exactly one repeated real root (parabola touches the X-axis at its vertex).
• Δ < 0: The equation has two complex conjugate roots (parabola does not intersect the X-axis).

4. Worked Example: Step by Step

Solve 2x² − 4x − 6 = 0:

1. Identify coefficients: a = 2, b = −4, c = −6.
2. Compute discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64.
3. Since Δ = 64 > 0, there are two real roots: √64 = 8.
4. Apply formula: x = (4 ± 8) / (2 × 2) = (4 ± 8) / 4.
5. x₁ = (4 + 8) / 4 = 12 / 4 = 3; x₂ = (4 − 8) / 4 = −4 / 4 = −1.