It solves four kinds of equation and shows how: a linear equation ax + b = 0, a quadratic ax² + bx + c = 0, a cubic ax³ + bx² + cx + d = 0, and a system of two or three linear equations in x, y and z. You type the coefficients; the answer and the working update as you type.
Coefficients are read as exact fractions, not as floating-point numbers. Type 0.1 and the solver works with 1/10; type 2/3 and it works with two thirds. That is why the answers come out the way a textbook writes them: x = −1 ± √3 rather than 0.7320508076 and −2.732050808 (both decimals are shown as well), x = −1/2 rather than −0.5000000001, and a system answer of x = 5/7 rather than a rounded decimal.
How each kind is solved
Quadratic. The discriminant Δ = b² − 4ac decides everything. For x² + 2x − 2 = 0, Δ = 4 + 8 = 12. Twelve is 4 × 3, so √12 = 2√3, and the formula x = (−b ± √Δ)/(2a) gives (−2 ± 2√3)/2 = −1 ± √3. When Δ is a perfect square the roots are fractions and the tool also writes the factored form: x² − 5x + 6 = (x − 3)(x − 2). When Δ = 0 there is one double root (4x² + 4x + 1 = 0 gives x = −1/2). When Δ is negative the roots are complex: x² + 2x + 5 = 0 has Δ = −16 and roots −1 ± 2i. You also get the vertex of the parabola and a check that the roots add up to −b/a and multiply to c/a. The article what the discriminant tells you works through more examples.
Cubic. First the rational root test: any fraction that solves a cubic with whole coefficients must be ±p/q, with p dividing the constant and q dividing the leading coefficient. For x³ − 6x² + 11x − 6 the candidates are ±1, ±2, ±3, ±6; x = 1 works, dividing it out leaves x² − 5x + 6, and the roots are 1, 2 and 3. When no candidate works the roots are irrational and are found numerically: the trigonometric method when the cubic discriminant is positive (three real roots), Cardano's formula when it is negative (one real root and a complex pair). x³ − 2 = 0 gives ∛2 ≈ 1.25992105 and −0.6299605249 ± 1.091123636i. Every numerical answer is substituted back into the equation and the largest leftover is shown, so you can see the roots really work.
Systems. The equations become an augmented matrix and the tool runs Gauss–Jordan elimination with exact fractions, listing every row operation and the matrix after it. It reports one solution, no solution (a row that says 0 = 1) or infinitely many, written with a parameter: x + 2y = 4 and 2x + 4y = 8 give x = 4 − 2t, y = t.
Limits worth knowing
Roots are exact whenever the algebra allows: fractions, square roots and cube roots of a single number. A cubic like x³ − 3x + 1 = 0 has three real roots that cannot be written with real square and cube roots alone (the "casus irreducibilis"), so they are shown as decimals to ten significant digits. Square roots are simplified by pulling out square factors below 100 000, which covers anything typed by hand. Quartic and higher equations are not solved here; to see where any function crosses zero, plot it with the function grapher. For larger linear systems, determinants and inverses, use the matrix calculator, and for plain arithmetic the scientific calculator.
