THE NUMORIX GUIDE
How to use the Quadratic Equation Calculator
Last reviewed September 14, 2026
What this calculator does
For ax^2+bx+c=0, calculate discriminant D=b^2-4ac, then roots x=(-b +/- sqrt(D))/(2a) when D is nonnegative; negative D produces a conjugate complex pair.
Formula and method
For ax^2+bx+c=0, calculate discriminant D=b^2-4ac, then roots x=(-b +/- sqrt(D))/(2a) when D is nonnegative; negative D produces a conjugate complex pair. The vertex is ( -b/(2a), f(-b/(2a)) ).
Variables and inputs
a, b, and c are coefficients; the visible defaults are a=1, b=-5, c=6. The UI requires a nonzero a for the quadratic path.
Worked example
For x^2-5x+6=0: D = (-5)^2 - 4*1*6 = 1; roots = (5 +/- 1)/2 = 3 and 2; vertex x = 5/2 = 2.5 and y = 2.5^2 - 5*2.5 + 6 = -0.25.
How to interpret the result
The discriminant tells how many real roots exist: positive means two, zero means one repeated root, and negative means no real roots but two complex roots.
Common mistakes to avoid
Keep the sign of b inside -b and use 2a as the entire denominator. Do not call complex roots real solutions.
Assumptions and limitations
The engine rounds root strings to four decimals and does not rationalize radicals. If a is zero, it falls back to a linear calculation internally, although the visible UI blocks that input.
Practical use and checks
Use the Quadratic Equation Calculator for equations in the form ax^2 + bx + c = 0. Enter a = 1, b = -5, and c = 6 as a check; the roots should be 2 and 3 because x^2 - 5x + 6 factors as (x - 2)(x - 3). The discriminant is 1, so there are two real roots, and the vertex should be at x = 2.5 with y = -0.25. Substituting either displayed root back into the original expression should give approximately zero. The discriminant is a useful decision point: positive gives two real roots, zero gives one repeated root, and negative gives a complex pair. Keep the sign on b inside -b and use 2a as the full denominator. A zero value of a changes the problem into a linear equation; the visible form may block it, so do not call a linear result a quadratic solution. The calculator rounds root strings and does not preserve exact radicals, which can matter when roots are nearly repeated or coefficients are large. It also assumes real-number coefficients and does not model inequalities, domain restrictions, or the meaning of a root in an applied problem.