Quadratic Equation Solver
Every quadratic ax² + bx + c = 0 has a fixed formula for its roots — the trick is the discriminant, which tells you in advance whether those roots are two real numbers, one repeated real number, or a complex pair, before you finish the arithmetic.
Inputs
Result
x₁ = 2, x₂ = 1
How it works
x = (−b ± √(b² − 4ac)) ÷ 2a
- 1Discriminant = 1
- 2x = (−-3 ± √1) ÷ 2
How the quadratic equation solver works
The quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
The discriminant, b² − 4ac, decides the outcome: positive gives two distinct real roots, exactly zero gives one repeated real root, and negative gives two complex roots.
When the discriminant is negative, the square root of a negative number becomes an imaginary component, and the two roots are complex conjugates of each other.
Worked example: x² − 3x + 2 = 0
- Discriminant = (−3)² − 4×1×2 = 9 − 8 = 1.
- Since 1 > 0, expect two distinct real roots.
- x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
- x₁ = 4 ÷ 2 = 2, x₂ = 2 ÷ 2 = 1.
Common mistakes to avoid
Forgetting the ± produces two separate answers
A quadratic generically has two roots, not one — dropping the minus branch of ± silently discards half the solution.
Assuming a negative discriminant means an error
It means the roots are complex, not real — a legitimate mathematical outcome, not a broken input. It happens whenever the parabola never crosses the x-axis.
Entering a = 0
With no x² term, the equation is linear, not quadratic — use the Algebra Solver instead, since dividing by 2a with a = 0 is undefined.
Frequently asked questions
What does it mean when the discriminant is exactly zero?
The parabola touches the x-axis at exactly one point — a repeated root, meaning both x₁ and x₂ come out equal.
How do complex roots show up in the result?
As a real part plus or minus an imaginary part, e.g. x = 2 ± 3i, where i represents the square root of −1.
Can I verify a root without redoing the formula?
Substitute it back into ax² + bx + c — if the equation is genuinely satisfied, the result should equal exactly 0.
Does the order of a, b, c matter?
Yes — they must correspond exactly to the coefficients of x², x, and the constant respectively in that order.
Learn more
The Discriminant: Knowing the Answer's Shape Before You Solve
b² − 4ac tells you whether a quadratic has two real roots, one, or none — before you finish the formula.
Related calculators
Algebra Solver
Solve linear equations of the form ax + b = c.
Polynomial Calculator
Evaluate a cubic polynomial at any value of x.
Equation Solver
Solve simultaneous linear equations in two variables.
Exponent Calculator
Raise any base to any power.
Root Calculator
Square, cube and nth roots.