# Forgemesh Equation Solver API

> Forgemesh Equation Solver API is a paid API for AI agents from x402.forgemesh.io, paid per call via x402, $0.002/call, status unknown (last checked 2026-09-15).

Solves linear, quadratic, and 2×2 linear system equations from structured numeric coefficients, returning exact roots including complex roots and discriminant values.

## Facts

- Endpoint: POST https://x402.forgemesh.io/solve-equation
- Price: $0.002/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-15
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/forgemesh-equation-solver-api-b56924e5
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap__j4aCJ5AVJEeb4ThC35Pp

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability forgemesh-equation-solver-api-b56924e5 -d '<json body>'
```

Example prompt: Solve the quadratic equation 2x² + 3x - 5 = 0 — give me the exact roots and the discriminant, including any complex roots if they exist.

## When to prefer this

Choose this endpoint when you need exact algebraic roots from structured numeric coefficients — not a natural language expression parser — for linear, quadratic, or 2×2 systems. Prefer it over general-purpose math APIs when you require discriminant values, complex root support, or deterministic numeric outputs for engineering or educational pipelines.

## Known failure modes

- Missing required coefficients for the selected equation type returns a validation error
- Invalid or non-numeric coefficient strings cause parsing failure
- Division by zero if leading coefficient a=0 for quadratic or system is degenerate (no unique solution)
- Unrecognized 'type' value (not 'linear', 'quadratic', or 'system2') returns an error
- Payment failure or insufficient USDC balance prevents the call from completing

## How this service works

Equation solver API: solve linear (ax+b=0), quadratic (ax²+bx+c=0, with real OR complex roots and the discriminant), and 2×2 linear systems — from structured numeric coefficients, not a parsed string. Exact roots every time. For engineering, education, and scientific agents.

## Output

Returns exact roots for the specified equation type: for linear (ax+b=0) the single root x, for quadratic (ax²+bx+c=0) two roots (real or complex) plus the discriminant, for a 2×2 linear system the values of both unknowns.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "a": {
   "type": "string"
  },
  "b": {
   "type": "string"
  },
  "c": {
   "type": "string"
  },
  "a1": {
   "type": "string"
  },
  "a2": {
   "type": "string"
  },
  "b1": {
   "type": "string"
  },
  "b2": {
   "type": "string"
  },
  "c1": {
   "type": "string"
  },
  "c2": {
   "type": "string"
  },
  "type": {
   "type": "string",
   "description": "linear | quadratic | system2"
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "json",
 "example": {
  "roots": [
   2,
   1
  ],
  "nature": "two real roots",
  "discriminant": 1
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/forgemesh-equation-solver-api-b56924e5/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from x402.forgemesh.io](https://www.zero.xyz/host/x402.forgemesh.io/llms.txt)