Quadratic Equation Solver
Roots, discriminant and vertex for ax² + bx + c = 0, including complex roots.
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Your details
The x² coefficient.
The x coefficient.
The constant term.
Result
- Roots
- 2 and 1
- Discriminant (b² − 4ac)
- 1.0000
- Vertex
- (1.5000, -0.2500)
- Axis of symmetry
- x = 1.5000
- Opens
- Upward
two real roots
Positive — the parabola crosses the x-axis twice
Let's work it out…
x = (−b ± √(b² − 4ac)) ÷ 2a
x = (−(-3) ± √((-3)² − 4 × 1 × 2)) ÷ (2 × 1)
x = (3 ± √1) ÷ 2
x = 2 and 1
Formula — written step by step
Solving x² − 3x + 2 = 0…
x = (−b ± √(b² − 4ac)) ÷ 2a
x = (3 ± √(9 − 8)) ÷ 2
x = (3 ± 1) ÷ 2
x = 2 and x = 1
a, b and c are the coefficients of x², x and the constant. The ± is what produces two roots: one from adding the square root, one from subtracting it.
A worked example
A quadratic with no real roots
- a
- 1
- b
- 0
- c
- 1
- Discriminant = 0² − 4 × 1 × 1
- = −4
- A negative discriminant means the square root is imaginary
x = 0 + 1i and x = 0 − 1i
The parabola sits entirely above the x-axis, so it never crosses it. The roots still exist — they are just complex rather than real.
How to use this calculator
- Enter the three coefficients from ax² + bx + c = 0. Write the equation with everything on one side first.
- Read the discriminant to know what to expect: positive gives two roots, zero gives one, negative gives a complex pair.
- The vertex and axis of symmetry describe the parabola itself, which is what you need to sketch it.
- Setting a to zero is allowed — the equation becomes linear and the single solution is shown instead.
The discriminant tells you the answer before you find it
The quantity b² − 4ac sits under the square root, and its sign alone decides the shape of the answer. Positive means the root is a real number and the ± gives two distinct solutions. Zero means the ± adds and subtracts nothing, so the two roots coincide. Negative means the square root is imaginary and the solutions are a complex conjugate pair.
Geometrically it is where the parabola sits relative to the x-axis: crossing it twice, touching it once, or missing it entirely.
Complex roots are answers, not failures
A negative discriminant is often taught as "no solution", which is only true if you have restricted yourself to real numbers. The roots exist as complex numbers, and they matter — they describe oscillation in electrical circuits and control systems, where a complex pair means the system rings rather than settling.
They always arrive in conjugate pairs when the coefficients are real: if 2 + 3i is a root, so is 2 − 3i.
Frequently asked questions
What if a is zero?
Then there is no x² term and the equation is linear, not quadratic. The single solution −c/b is shown instead, and the formula does not apply because it would divide by zero.
What does the vertex tell me?
It is the turning point of the parabola — the minimum when a is positive, the maximum when a is negative. It is the answer to optimisation questions like maximum profit or peak height.
Can I always factorise instead?
Only when the roots are rational. The formula works for every quadratic, which is why it is worth knowing even when factorising is quicker for tidy examples.