Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots, with the discriminant and vertex.
Roots
x = 1, x = 2
- Equation
- x² − 3x + 2 = 0
- Discriminant (b² − 4ac)
- 1
- Vertex
- (1.5, -0.25)
- Sum and product of roots
- 3 and 2
Show the work
Discriminant
D = b to the power of 2 − 4ac
(−3) to the power of 2 − 4 × 1 × 2 = 1
D > 0, so there are two real roots
x = (−b ± √D) ÷ 2a
(−(−3) ± √1) ÷ (2 × 1) = 1 and 2
The roots
x = 1, x = 2
How it works
x = (−b ± √(b² − 4ac)) ÷ 2a; discriminant D = b² − 4ac (D < 0 → complex roots)
Questions
- How does the quadratic equation solver work?
- x = (−b ± √(b² − 4ac)) ÷ 2a; discriminant D = b² − 4ac (D < 0 → complex roots).
- What do I need to enter?
- a, b, c. Optional fields can be left blank.
Related tools
- Cubic Equation SolverSolve ax³ + bx² + cx + d = 0 for all three roots, real or complex.
- FOIL CalculatorMultiply two binomials (ax + b)(cx + d) with the FOIL method and see each step.
- Completing the Square CalculatorRewrite ax² + bx + c in vertex form a(x − h)² + k by completing the square.
- Fraction, Decimal & Percent ConverterConvert between fractions, decimals and percents — type any one, get all three, including repeating decimals.
- Ratio to Fraction CalculatorTurn a ratio like 3 : 5 into fractions of the whole, percentages and the simplified ratio.
- Percent Error CalculatorFind the percent error between a measured (experimental) value and the true (accepted) value.
Want this on your site? Embed the quadratic equation solver.