Quadratic Formula Calculator
Enter a, b, and c from ax² + bx + c = 0 and get both roots — real or complex — plus the discriminant, the vertex of the parabola, a factored form when one exists over the rationals, and the full substitution steps.
Reading the results
The discriminant b² − 4ac decides everything: positive gives two real crossings of the x-axis, zero gives a single repeated root (the parabola just touches the axis), and negative gives a complex-conjugate pair (no crossing). The vertex is the turning point — minimum if the parabola opens upward (a > 0), maximum if downward. When the coefficients are integers and the discriminant is a perfect square, the quadratic also factors neatly, and the calculator writes that factored form out.
How it’s calculated
x = (−b ± √(b² − 4ac)) ÷ 2a. Vertex: h = −b/2a, k = c − b²/4a. Complex case: x = −b/2a ± (√|D|/2a)i. Factored form a(x − r₁)(x − r₂) is printed when a, b, c are integers and D is a perfect square, with roots reduced to lowest-term fractions.
If a = 0 the equation is linear, not quadratic — the calculator then solves bx + c = 0 directly and says so.
Worked example
x² − 5x + 6 = 0: the discriminant is (−5)² − 4(1)(6) = 1, so x = (5 ± 1)/2, giving roots 3 and 2 and factored form (x − 2)(x − 3). The vertex sits at (2.5, −0.25). A complex case: 2x² + 2x + 3 = 0 has discriminant −20, so x = −0.5 ± 1.118034i.
Common mistakes
- Dropping the sign of b: with b = −5, −b is +5, and squaring gives 25 either way.
- Dividing only the radical by 2a instead of the whole numerator (−b ± √D).
- Forgetting the equation must equal zero first — move every term to one side before reading off a, b, c.
- Declaring “no solution” for a negative discriminant when the question expects complex roots.
Where it is used
- Algebra homework: solving, factoring, and graphing quadratics.
- Physics: projectile motion — when does height h(t) = 0?
- Business: break-even and profit-maximization models with squared terms.
- Engineering: characteristic equations of second-order systems.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. Substituting a, b, and c and simplifying the square root gives both roots at once — the calculator shows that substitution line by line.
What does the discriminant tell me?
The discriminant is b² − 4ac. Positive means two distinct real roots, zero means one repeated real root, and negative means a complex-conjugate pair. It also flags factorability: with integer coefficients, a perfect-square discriminant means the quadratic factors over the rationals.
What are complex roots?
When b² − 4ac is negative, the square root involves i = √−1, giving roots of the form p ± qi. For 2x² + 2x + 3 = 0 the discriminant is −20, so x = −0.5 ± 1.118i. The parabola never crosses the x-axis in that case.
What is the vertex and why does it matter?
The vertex (−b/2a, c − b²/4a) is the parabola’s turning point — the minimum when a > 0 or maximum when a < 0. It is the key to graphing the quadratic and to optimization problems like maximum height or minimum cost.
Why does my textbook answer look different from the decimal roots?
Textbooks often leave answers in exact radical form like (5 ± √17)/4, while calculators show decimals. Both are the same numbers; check the steps line, which keeps the radical form before converting.