Quadratic equation solver
Type the equation the way it's written on the page — something like 2x^2 - 7x + 12 = 0 — and the coefficients fill in automatically, or enter a, b and c directly if you already have them. Either way you get the roots, the discriminant, the vertex and the factored form.
How to use the quadratic equation solver
- Type the equation in expanded form, or enter a, b and c directly if you already have them.
- Check the coefficients the tool read back — adjust them if the equation was ambiguous.
- Read the roots and the discriminant, then the vertex and axis of symmetry if you're sketching the curve.
How it works
Typing an equation works by reading it left to right, splitting on the equals sign, and moving every term to one side — so 3x - 5 = 2x + 7 becomes x - 12 = 0 before solving. This only handles equations already in expanded form; something like 2(x-3)^2 = 8 needs multiplying out by hand first, since parsing unexpanded brackets correctly needs a much larger algebra engine than a quick equation reader can reliably manage.
The roots come from the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. The expression under the square root, b² − 4ac, is the discriminant, and it decides everything: positive gives two distinct real roots, zero gives one repeated root where the curve touches the axis, and negative gives a conjugate pair of complex roots because a negative number has no real square root.
The vertex sits at x = −b ÷ 2a, halfway between the roots when they exist, and the sign of a tells you whether it is the minimum or the maximum of the curve. Two useful checks come free from Viète's formulas: the roots must sum to −b/a and multiply to c/a.
Common questions
Can I just type my equation instead of finding a, b and c myself?
Yes — type it in expanded form (like 3x^2 - 4x - 2 = 0) and the tool reads off the coefficients for you, showing what it understood so you can correct it if needed.
Why doesn't it accept brackets, like 2(x-3)^2 = 8?
Expanding brackets correctly needs real algebra rather than pattern-matching text, and getting it subtly wrong would be worse than not supporting it — multiply it out first and paste the expanded form instead.
What does the discriminant tell me?
How many real roots there are. Positive means two, zero means one repeated root, negative means no real roots and a complex pair instead.
What if a is zero?
Then it is not quadratic — it is a straight line, and the tool solves it as one.
Why are my roots imaginary?
Because the parabola never crosses the x-axis. The complex roots still exist and are shown in a ± bi form.