Quadratic equation solver
Solve an equation of the form ax² + bx + c = 0
x² +
x +
= 0
❓ How the calculator works
The calculator solves a quadratic equation of the form ax² + bx + c = 0. First it computes the discriminant D, then determines the roots by its sign. Coefficient a cannot be zero — otherwise the equation is not quadratic.
📐 Formulas
Discriminant
D = b² − 4ac
Roots
x = (−b ± √D) / (2a)
When D = 0: x = −b / (2a)
- D > 0 — two distinct real roots
- D = 0 — one root (a double root)
- D < 0 — no real roots
💡 Calculation example
Equation x² + 5x + 6 = 0 (a = 1, b = 5, c = 6):
- D = 5² − 4 × 1 × 6 = 25 − 24 = 1
- x₁ = (−5 + 1) / 2 = −2
- x₂ = (−5 − 1) / 2 = −3
Check: (x + 2)(x + 3) = 0 — the roots are correct.
✅ Tips
- Simplify the equation first: divide all terms by a common divisor if there is one.
- If D is a perfect square, the roots have no fractional part.
- For D < 0, advanced courses use complex roots: x = (−b ± i√|D|) / (2a).
- Vieta’s formulas help verify the result: for x² + bx + c = 0 the sum of roots is −b and the product is c.