Quadratic Formula Calculator
Enter the coefficients a, b and c of a quadratic equation to get both roots via the quadratic formula, with the discriminant case analysis (two real, one repeated, or complex conjugates), the vertex and a sketch of the parabola.
TheCalculators — Quadratic Formula Calculator
Printed thecalculators.com
Roots of 1x² − 3x + 2 = 0
x₁ = 2, x₂ = 1
Discriminant D = 1 · D > 0 → two distinct real roots
- Coefficients (a, b, c)
- 1, -3, 2
- Discriminant b² − 4ac
- 1
- Case
- D > 0 → two distinct real roots
- √D
- 1
- Sum of roots (−b/a)
- 3
- Product of roots (c/a)
- 2
- Vertex
- (1.5, -0.25)
- Axis of symmetry
- x = 1.5
- Opens
- Upward (minimum)
Discriminant case analysis
- 1Identify coefficientsa = 1, b = -3, c = 2
- 2Compute the discriminantD = b² − 4ac = (-3)² − 4·1·2 = 9 − 8 = 1
- 3Interpret DD > 0 → two distinct real roots
- 4Apply the quadratic formulax = (−b ± √D)/(2a) = (3 ± 1) / 2 → x₁ = 2, x₂ = 1
- 5Vertex (completing the square)x = −b/(2a) = 1.5 · f(vertex) = -0.25
Parabola y = ax² + bx + c
Most useful next
Formula
x = (−b ± √(b² − 4ac)) / 2a · D = b² − 4ac
- • D > 0: two distinct real roots
- • D = 0: one repeated real root x = −b/2a
- • D < 0: complex roots x = (−b ± i√(4ac − b²))/2a
Pro tips
- • The discriminant alone tells you how many real solutions exist before solving.
- • Complex roots always come in conjugate pairs p ± qi when a, b, c are real.
- • Factoring first is often faster when the roots are nice integers.
FAQ
Related Calculators
More Everyday tools you might need.
Scientific Calculator
Full scientific keypad — sin, cos, tan, logs, roots, powers, memory and Deg/Rad, with instant exact results.
CalculateExponent Calculator
Raise any base to any power — integer, negative or fractional — with a growth chart.
CalculateRoot Calculator
Any nth root of any number — square, cube or higher — with even-root-of-negative guard.
Calculate