Formula & Calculator
Quadratic Formula
Solves any quadratic equation of the form ax^2 + bx + c = 0 directly from its coefficients.
Interpretation
x = (−b ± √(b²−4ac))/(2a). Solves ax²+bx+c=0. Finds roots of quadratic equation. Fundamental for algebra and many applications.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| a, b, c | Coefficients of the quadratic equation | |
| x | Root(s) of the equation |
What it means
The quadratic formula gives the solutions (roots) to any quadratic equation ax² + bx + c = 0, where a ≠ 0. The discriminant Δ = b² − 4ac determines the nature of the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, two complex roots. This formula is indispensable in algebra, physics (projectile motion), engineering (circuit analysis), and many other fields. It is derived by completing the square. Understanding the quadratic formula is a core requirement for high school and college mathematics, and it is one of the most frequently used formulas in problem solving.
Worked example
Quadratic Formula – Two Examples
Real‑World| Parameter | Value |
|---|---|
| a | −5 |
| b | 20 |
| c | 2 |
| Parameter | Value |
|---|---|
| a | 1 |
| b | 3 |
| c | −50 |
Common mistakes
- Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a) – not just −b/2a.
- Discriminant: D = b² − 4ac – determines number of real roots.
- If D > 0: Two distinct real roots; D = 0: one double root; D < 0: two complex roots.
- Sign: The ± means two possible values – include both.
- a ≠ 0: If a = 0, the equation is linear – use a different method.
Applications
The quadratic formula, x = (−b ± √(b² − 4ac))/(2a), provides the roots of a quadratic equation ax²+bx+c=0. This is one of the most important algebraic formulas, used to solve a vast array of problems. In physics, it appears in projectile motion and energy conservation. In engineering, it solves for time constants, resonant frequencies, and optimises designs. In finance, it calculates break‑even points. In computer science, it is used in graphics and collision detection. By understanding the discriminant, professionals can determine the nature of solutions (real, repeated, complex). This formula is indispensable in any field that involves polynomial equations.
- Projectile motion and ballistics calculations
- Circuit analysis (impedance matching and resonant circuits)
- Optimisation of quadratic cost functions in economics
- Root‑finding in computational geometry and ray tracing
- Stability analysis of second‑order systems
Frequently Asked Questions
The quadratic formula solves for the roots of a quadratic equation ax² + bx + c = 0: x = (–b ± √(b² – 4ac)) / (2a). It gives the values of x that satisfy the equation.
Forgetting the plus‑or‑minus sign, which produces only one of the two roots. The ± is essential.
The discriminant is Δ = b² – 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root (repeated); if Δ < 0, two complex roots.
For ax² + bx + c = 0, the sum of roots is –b/a, and the product is c/a.
By completing the square on ax² + bx + c = 0: divide by a, move the constant, add (b/(2a))², and solve for x.
- Solving projectile motion equations.
- Finding intersection points of parabolas.
- Optimization problems.
The vertex is at x = –b/(2a). The y‑coordinate is found by substituting this into the equation.
Use the quadratic formula; the square root of a negative number gives imaginary numbers. For example, x = (–b ± i√(|Δ|))/(2a).
If a = 0, the equation is not quadratic but linear. The formula is invalid (division by zero).
Factoring is an alternative method; the formula works for any quadratic, even when factoring is difficult.