Mathematical Tools
What It Does
Solves linear, quadratic, and cubic equations and shows the work — the discriminant, the formula with your numbers plugged in, and every step — plus real and complex roots and (for quadratics) the vertex and a small graph. Everything runs in your browser.
How to Use It
- Choose the equation type (linear, quadratic, or cubic).
- Enter the coefficients (a, b, c, d as needed).
- Read the roots and the step-by-step working; toggle the graph.
- Copy any root.
Options Explained
| Option | Description |
|---|---|
| Type | Linear (ax+b=0), quadratic (ax²+bx+c=0), or cubic (ax³+bx²+cx+d=0). |
| a, b, c, d | The coefficients; the leading one (a) must be non-zero. |
| Precision | How many significant digits to show. |
| Steps | Show the full worked solution (discriminant, formula, substitutions). |
| Graph | Show a small sketch of the curve with its real roots marked. |
Enter the coefficients to solve the equation.
About the Discriminant, Complex Roots & the Vertex
For a quadratic ax² + bx + c = 0, the discriminant Δ = b² − 4ac is the quantity under the square root in the quadratic formula, and its sign alone tells you what the roots look like: a positive Δ gives two distinct real roots (the parabola crosses the x-axis twice), a zero Δ gives one repeated real root (it just touches the axis), and a negative Δ gives a complex-conjugate pair a ± bi (it never touches the axis).
Complex roots always come in conjugate pairs for real-coefficient polynomials, which is why an “imaginary” answer isn’t an error — it’s a perfectly valid solution in the complex plane. Every parabola also has a vertex, its highest or lowest point, at x = −b/2a; the vertical line through it, the axis of symmetry, mirrors the two roots around it.
Cubics are trickier: they always have at least one real root, and this tool depresses the cubic and uses either a trigonometric formula (three real roots) or Cardano’s formula (one real root plus a complex pair) depending on the cubic discriminant. Because floating-point arithmetic can turn a true repeated root into a tiny imaginary sliver, the solver rounds a near-zero discriminant to exactly zero — and, like every tool here, it runs entirely on your device.