Quadratic equation solver

Graph

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Formula

How it works

The discriminant b² − 4ac tells you the kind of roots. If it is positive there are two real roots, if zero one repeated root, and if negative two complex roots. The quadratic formula gives their values.

Worked example

The same calculation with the default values:

• Coefficient a (of x²) 1
• Coefficient b (of x) -3
• Constant c 2

1. b² − 4ac = (-3)² − 4 × (1) × (2) = 1
2. √1 = 1
3. x₁ = (−(-3) + 1) ÷ (2 × 1) = 2
4. x₂ = (−(-3) − 1) ÷ (2 × 1) = 1

Result: Discriminant (b² − 4ac): 1 · Root x₁: 2 · Root x₂: 1

Common mistakes

• Forgetting the ± in the quadratic formula - there are generally two roots, not one; dropping the minus branch silently discards a valid solution.
• Making a sign error with b or substituting -b incorrectly - the formula uses negative b, so a positive b in the equation becomes negative inside the formula, and this sign is easy to drop by mistake.
• Assuming every quadratic has two real-number solutions - when the discriminant is negative, the solutions are complex, which is a mathematically valid but non-real result, not a sign of an error in the input.