How to Calculate Quadratic Formula
What is Quadratic Formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x. It works for all quadratics — even ones that cannot be factored — making it the most universal solving method. The formula was known to Babylonian mathematicians as early as 2000 BC.
Step-by-Step Guide
- 1Arrange the equation in standard form: ax² + bx + c = 0
- 2Identify a (coefficient of x²), b (coefficient of x), c (constant)
- 3Calculate the discriminant: Δ = b² − 4ac
- 4If Δ ≥ 0: substitute into x = (−b ± √Δ) / 2a for two real roots
- 5If Δ < 0: the equation has two complex (non-real) roots
Worked Examples
Input
x² − 5x + 6 = 0
Result
x = 3 or x = 2
Δ = 25−24 = 1 > 0. Roots: (5±1)/2
Input
x² − 2x + 1 = 0
Result
x = 1 (repeated)
Δ = 4−4 = 0. One repeated root.
Input
x² + x + 1 = 0
Result
Complex roots
Δ = 1−4 = −3 < 0. No real solutions.
Input
2x² + 3x − 2 = 0
Result
x = 0.5 or x = −2
Δ = 9+16 = 25. Roots: (−3±5)/4
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