Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

The most reliable method that works for all quadratic equations. The discriminant (b² - 4ac) determines the number of real solutions.

The Discriminant

• If b² - 4ac > 0: Two distinct real solutions
• If b² - 4ac = 0: One repeated real solution
• If b² - 4ac < 0: No real solutions (complex solutions)

Factoring Method

When possible, factor the equation into (x - r₁)(x - r₂) = 0

Example: x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3

Completing the Square

Convert to the form (x - h)² = k and solve for x

Steps:

1. Move constant to right side
2. Complete the square on left side
3. Take square root of both sides
4. Solve for x

Projectile Motion

Calculate the trajectory of objects thrown or launched, finding maximum height and landing distance.

Business & Economics

Find profit maximization points, break-even analysis, and optimal pricing strategies.

Engineering

Design bridges, arches, and structures; analyze stress and load distributions.

Area Problems

Calculate dimensions for maximum area with given perimeter constraints in construction and landscaping.