Equation Solver
Solves linear, quadratic and cubic equations with the working shown, finds real roots of other equations numerically, and solves linear systems by elimination.
Solves an equation in one unknown, or a system of linear equations, with the working shown. Degrees one to three are solved exactly by formula; anything else — a quartic, something with a sine in it — is bracketed and refined numerically for its real roots.
Equation
Only used when there is no formula
Answer
Quadratic — the quadratic formula · rearranged: x^2 - 4x - 2 = 0
- x = -0.44949
- x = 4.44949
Working
1. Put it in standard form
x^2 - 4x - 2 = 0 (a = 1, b = -4, c = -2)
2. The discriminant
D = b² − 4ac = (-4)² − 4·(1)·(-2) = 24
3. The quadratic formula
x = (−b ± √D) / 2a = (4 ± 4.89898) / 2
4. Two real roots
x = -0.44949 or x = 4.44949
What you can write: + - * / ^, brackets, implicit multiplication such as 2x and 3(x+1), sin, cos, tan, exp, ln, log, sqrt and abs, and the constants pi and e. An unknown is a single letter (x, y, t, a…) or a letter with a number (x1, x2). e is the natural constant, so it cannot be an unknown.
Degrees 1, 2 and 3 are solved exactly by formula and the working is shown. Degree 4 and up, and anything containing a function, are handled by slicing the range, finding where the sign changes, narrowing by bisection and finishing with Newton — real roots only. Systems use Gaussian elimination with partial pivoting.
How to use it
Write it as you would on paper
Implicit multiplication works: 2x, 3(x+1) and x(x-1) all mean what they look like. The equals sign is where you put it.
Or one equation per line for a system
Any number of unknowns; they are picked up from the equations. Gaussian elimination with partial pivoting does the rest.
Widen the search range if nothing is found
For equations with no formula the roots are found by looking for sign changes in a range, which defaults to −100 to 100. A root at 500 is outside it until you say otherwise.
Common questions
Why are there no complex roots?
For quadratics and cubics they are given. For anything solved numerically only real roots are found, because the method looks for sign changes on the real line and a complex root does not cause one.
A root exists but was not found
Either it is outside the search range, or the curve touches the axis without crossing it — a double root of the kind x² has at zero. Bisection needs a sign change, and a tangent root does not give one.
Can I use e as an unknown?
No. Here e is the natural constant, as is pi. Any other single letter, or a letter with a number after it like x1, works as an unknown.
The working is in Korean
The step-by-step derivation is Korean only — its thirty step titles are asserted word for word in the solver’s tests. The answer, the method used and every warning are in your language; only the derivation panel is not.
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