The Quadratic Formula
For any equation ax² + bx + c = 0 where a ≠ 0, the solutions are given by:
x = (-b ± √(b² - 4ac)) / 2a
Discriminant (b² - 4ac)
The discriminant determines the nature of the roots:
- Positive: Two distinct real roots
- Zero: One repeated real root (parabola touches x-axis)
- Negative: Two complex conjugate roots (parabola doesn't cross x-axis)
Examples
- x² - 5x + 6 = 0 → x = 2 or x = 3 (discriminant = 1)
- x² - 4x + 4 = 0 → x = 2 (discriminant = 0)
- x² + x + 1 = 0 → x = -0.5 ± 0.866i (discriminant = -3)
Vertex Form
The vertex of the parabola y = ax² + bx + c is at (-b/2a, c - b²/4a). The axis of symmetry is x = -b/2a.
Other Methods for Solving Quadratics
Factoring
If the equation factors neatly, this is the fastest method. x² - 5x + 6 = 0 factors to (x-2)(x-3) = 0, giving x = 2 or x = 3. Not all quadratics factor into integers — the quadratic formula works for every case.
Completing the Square
Rearrange ax² + bx + c = 0 into vertex form a(x-h)² + k = 0. This method is how the quadratic formula itself is derived. It's useful when you need the vertex for graphing.
Real-World Applications
- Projectile motion: The height of a thrown ball follows h(t) = -16t² + v₀t + h₀ (in feet). Setting h = 0 and solving gives the time when it hits the ground
- Area problems: If a rectangle has perimeter 30 and area 50, the dimensions satisfy x² - 15x + 50 = 0, giving x = 5 or x = 10
- Revenue optimization: Revenue R = price × quantity, where quantity often depends linearly on price, creating a quadratic equation. The vertex gives the price that maximizes revenue
- Engineering: Structural load calculations, circuit analysis, and signal processing all use quadratic equations
Quick Tips
- If c = 0, the equation simplifies: ax² + bx = 0 → x(ax + b) = 0 → x = 0 or x = -b/a
- If b = 0, solve directly: ax² + c = 0 → x = ±√(-c/a)
- The sum of the roots equals -b/a and the product equals c/a (Vieta's formulas)
Graphing Quadratic Functions
The graph of y = ax² + bx + c is always a parabola. Key features you can determine algebraically:
- Direction: Opens upward if a > 0, downward if a < 0
- Vertex: The turning point at x = -b/(2a). This is the minimum (a > 0) or maximum (a < 0) value
- Axis of symmetry: The vertical line x = -b/(2a). The parabola is symmetric about this line
- Y-intercept: The constant term c (set x = 0)
- X-intercepts: The roots found by the quadratic formula (if real)
History of the Quadratic Formula
Babylonian mathematicians solved quadratic equations as early as 2000 BC, though they used geometric methods rather than a formula. The Indian mathematician Brahmagupta (628 AD) provided one of the earliest explicit solutions for the general quadratic equation. The formula as we know it today was formalized by European mathematicians during the Renaissance. Despite being over 1,000 years old, the quadratic formula remains one of the most frequently used tools in algebra, engineering, physics, and computer graphics.