About the quadratic formula calculator
The quadratic formula calculator solves any equation of the form ax² + bx + c = 0. It reports both roots, whether they are real or complex, along with the discriminant, the vertex, the axis of symmetry and the y-intercept of the parabola.
You can enter coefficients as whole numbers, decimals or fractions such as 3/4.
Who should use the quadratic formula calculator
- Algebra students solving or checking quadratics.
- Physics and engineering learners analysing projectile or optimisation problems.
- Teachers needing a quick worked solution.
How to use the quadratic formula calculator: what each input means
- a
- The coefficient of x². It cannot be zero, otherwise the equation is not quadratic.
- b
- The coefficient of x.
- c
- The constant term.
How the quadratic formula calculator works: method and formula
The discriminant b² − 4ac determines the type of roots: positive gives two real roots, zero gives one repeated root, and negative gives two complex roots. The roots are then found with the quadratic formula. The calculator uses a numerically stable form that avoids losing precision when b is large compared with a and c.
- Quadratic formula
- x = (−b ± √(b² − 4ac)) ÷ (2a)
- Discriminant
- Δ = b² − 4ac
- Vertex
- x = −b ÷ 2a, y = a·x² + b·x + c
How to read your quadratic formula calculator result
Real roots are where the parabola crosses the x-axis. A repeated root means it just touches the axis. Complex roots mean it never crosses it. The vertex is the minimum point if a > 0 and the maximum if a < 0.
Quadratic Formula Calculator worked example
Solve x² − 3x + 2 = 0.
- a = 1, b = −3, c = 2, so the discriminant is (−3)² − 4 × 1 × 2 = 1.
- x = (3 ± √1) ÷ 2, giving x = 1 and x = 2.
- The vertex is at x = 1.5, y = -0.25.
- For comparison, x² + 2x + 5 = 0 has discriminant -16 and complex roots -1 ± 2i.
x = 1 or x = 2.
Practical uses of the quadratic formula calculator
- Solving homework equations.
- Finding when a thrown object reaches the ground.
- Finding the maximum or minimum of a quadratic model.
Assumptions and limitations of this calculator
- Results are floating-point approximations and are displayed to 12 significant digits.
- The equation must be in standard form. Rearrange first if needed.