Quadratic Formula
The quadratic formula solves any equation of the form ax² + bx + c = 0.
Find both roots instantly with this essential algebra formula.
The Formula
x = (-b ± √(b² - 4ac)) / 2a
This formula gives the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0.
Variables
| Symbol | Meaning |
|---|---|
| x | The unknown variable (the solution you are solving for) |
| a | Coefficient of x² (must not be zero) |
| b | Coefficient of x |
| c | Constant term |
| b² - 4ac | The discriminant — determines the number of real solutions |
Understanding the Discriminant
- If b² - 4ac > 0 → two distinct real roots
- If b² - 4ac = 0 → one repeated real root
- If b² - 4ac < 0 → no real roots (two complex roots)
Example 1
Solve: 2x² + 5x - 3 = 0
Here a = 2, b = 5, c = -3
Discriminant: b² - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49
x = (-5 ± √49) / (2 × 2) = (-5 ± 7) / 4
x = (-5 + 7) / 4 = 0.5 or x = (-5 - 7) / 4 = -3
Example 2
Solve: x² - 6x + 9 = 0
Here a = 1, b = -6, c = 9
Discriminant: (-6)² - 4(1)(9) = 36 - 36 = 0
x = (6 ± √0) / 2 = 6 / 2
x = 3 (one repeated root)
When to Use It
Use the quadratic formula when:
- Factoring is difficult or not obvious
- You need exact solutions (not approximations)
- The equation cannot be easily simplified
- You want to determine whether real solutions exist (check the discriminant first)
Key Notes
- Formula: x = (−b ± √(b²−4ac)) / 2a: Solves any equation in the form ax² + bx + c = 0. The ± gives two solutions. The formula is derived by completing the square on the general quadratic — it is not a separate rule, but a general result.
- The discriminant b² − 4ac determines the nature of roots: Positive → two distinct real roots; zero → one repeated real root (the parabola is tangent to the x-axis); negative → two complex conjugate roots (the parabola does not cross the x-axis).
- Vertex x-coordinate is −b/2a: The axis of symmetry passes through x = −b/2a, which is the average of the two roots. Substituting this x back into the equation gives the minimum (a > 0) or maximum (a < 0) value of the parabola.
- Sum and product of roots: x₁ + x₂ = −b/a; x₁ × x₂ = c/a (Vieta's formulas). These shortcuts let you check solutions or construct a quadratic from its roots without fully applying the formula.
- Applications: The quadratic formula solves projectile motion (find time when height = 0), break-even analysis (revenue = cost), lens optics, circuit analysis, and any physical or economic model described by a parabolic relationship.