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math_quadratic_solver

Read-onlyIdempotent

Solve quadratic equations from coefficients, evaluate f(x), or build quadratics from roots. Returns discriminant, roots, vertex, axis of symmetry, and factored form.

Instructions

Quadratic Equation Solver. Solve a quadratic equation ax^2 + bx + c = 0 from its coefficients, evaluate f(x) = ax^2 + bx + c at a point, or build a monic quadratic from two roots; selected by the operation field. Solving returns the discriminant and its sign, real or complex-conjugate roots, root count, vertex, axis of symmetry, factored form when the roots are rational, and the Vieta sum/product of roots; it also handles the degenerate a = 0 (linear) cases. Use this for second-degree polynomial algebra; use math_scientific_calculator for general expression evaluation and math_matrix_calculator for linear algebra. Runs locally on the numbers you provide: read-only, non-destructive, contacts no external service, and is rate-limited (60 requests/minute for anonymous callers).

Input Schema

TableJSON Schema
NameRequiredDescriptionDefault
operationYesMode to run. solve needs a, b, c. evaluate needs a, b, c, x. fromRoots needs r1, r2.
aNoQuadratic (x^2) coefficient for solve and evaluate. May be 0, in which case the equation is treated as linear or degenerate. Required for solve and evaluate.
bNoLinear (x) coefficient for solve and evaluate. Required for solve and evaluate.
cNoConstant term for solve and evaluate. Required for solve and evaluate.
xNoPoint at which to evaluate f(x) = ax^2 + bx + c. Required for the evaluate operation only.
r1NoFirst root of the quadratic to construct. Required for the fromRoots operation only.
r2NoSecond root of the quadratic to construct. Required for the fromRoots operation only.

Output Schema

TableJSON Schema
NameRequiredDescriptionDefault
successNoWhether the calculation succeeded.
operationNoThe operation performed (solve, evaluate, or fromRoots).
resultNoOperation output. solve fields listed here; evaluate returns a, b, c, x, y, vertex, axis; fromRoots returns a, b, c, equation.

Schema Changelog

Changes observed during successful MCP inspections.

  1. Changed14 schema fields changedv0.5.1
    • addedInput schema / additionalProperties
      Added value: +false
    • addedInput schema / properties / a / description
      Added value: +"Quadratic (x^2) coefficient for solve and evaluate. May be 0, in which case the equation is treated as linear or degenerate. Required for solve and evaluate."
    • changedInput schema / properties / a / type
      Previous value: -"integer"New value: +[
+  "number",
+  "null"
+]
    • addedInput schema / properties / b / description
      Added value: +"Linear (x) coefficient for solve and evaluate. Required for solve and evaluate."
    • changedInput schema / properties / b / type
      Previous value: -"integer"New value: +[
+  "number",
+  "null"
+]
    • addedInput schema / properties / c / description
      Added value: +"Constant term for solve and evaluate. Required for solve and evaluate."
    • changedInput schema / properties / c / type
      Previous value: -"integer"New value: +[
+  "number",
+  "null"
+]
    • addedInput schema / properties / operation / description
      Added value: +"Mode to run. solve needs a, b, c. evaluate needs a, b, c, x. fromRoots needs r1, r2."
    • addedInput schema / properties / operation / enum
      Added value: +[
+  "solve",
+  "evaluate",
+  "fromRoots"
+]
    • addedInput schema / properties / r1
      Added value: +{
+  "description": "First root of the quadratic to construct. Required for the fromRoots operation only.",
+  "type": [
+    "number",
+    "null"
+  ]
+}
    • addedInput schema / properties / r2
      Added value: +{
+  "description": "Second root of the quadratic to construct. Required for the fromRoots operation only.",
+  "type": [
+    "number",
+    "null"
+  ]
+}
    • addedInput schema / properties / x
      Added value: +{
+  "description": "Point at which to evaluate f(x) = ax^2 + bx + c. Required for the evaluate operation only.",
+  "type": [
+    "number",
+    "null"
+  ]
+}
    • changedInput schema / required
      Previous value: -[
-  "operation",
-  "a",
-  "b",
-  "c"
-]New value: +[
+  "operation"
+]
    • changedOutput schema / (root)
      Previous value: -nullNew value: +{
+  "properties": {
+    "operation": {
+      "description": "The operation performed (solve, evaluate, or fromRoots).",
+      "type": "string"
+    },
+    "result": {
+      "description": "Operation output. solve fields listed here; evaluate returns a, b, c, x, y, vertex, axis; fromRoots returns a, b, c, equation.",
+      "properties": {
+        "a": {
+          "description": "Rounded x^2 coefficient echoed back.",
+          "type": "number"
+        },
+        "axisOfSymmetry": {
+          "description": "Axis of symmetry x = -b/(2a), or null when not a genuine quadratic.",
+          "type": [
+            "number",
+            "null"
+          ]
+        },
+        "b": {
+          "description": "Rounded x coefficient echoed back.",
+          "type": "number"
+        },
+        "c": {
+          "description": "Rounded constant term echoed back.",
+          "type": "number"
+        },
+        "discriminant": {
+          "description": "b^2 - 4ac, rounded to 10 decimals.",
+          "type": "number"
+        },
+        "discriminantSign": {
+          "description": "Sign of the discriminant.",
+          "enum": [
+            "positive",
+            "zero",
+            "negative"
+          ],
+          "type": "string"
+        },
+        "equation": {
+          "description": "The cleaned equation string, for example x^2 - 3x + 2 = 0.",
+          "type": "string"
+        },
+        "factored": {
+          "description": "Factored form when both roots are rational, else null.",
+          "type": [
+            "string",
+            "null"
+          ]
+        },
+        "isDegenerate": {
+          "description": "True when a = 0 and b = 0 (no or infinite solutions).",
+          "type": "boolean"
+        },
+        "isLinear": {
+          "description": "True when a = 0 and b is non-zero (single linear root).",
+          "type": "boolean"
+        },
+        "productOfRoots": {
+          "description": "Vieta product c/a, or null when not a genuine quadratic.",
+          "type": [
+            "number",
+            "null"
+          ]
+        },
+        "rootCount": {
+          "description": "Number of distinct roots, or the string infinite for the 0 = 0 case.",
+          "type": [
+            "integer",
+            "string"
+          ]
+        },
+        "roots": {
+          "description": "Roots as objects with real, imag, and a display repr. Empty when there are zero or infinite solutions.",
+          "items": {
+            "properties": {
+              "imag": {
+                "description": "Imaginary part of the root (0 for real roots).",
+                "type": "number"
+              },
+              "real": {
+                "description": "Real part of the root.",
+                "type": "number"
+              },
+              "repr": {
+                "description": "Display string for the root.",
+                "type": "string"
+              }
+            },
+            "type": "object"
+          },
+          "type": "array"
+        },
+        "sumOfRoots": {
+          "description": "Vieta sum -b/a, or null when not a genuine quadratic.",
+          "type": [
+            "number",
+            "null"
+          ]
+        },
+        "vertex": {
+          "description": "Parabola vertex with x and y, or null for linear/degenerate cases.",
+          "properties": {
+            "x": {
+              "description": "Vertex x-coordinate.",
+              "type": "number"
+            },
+            "y": {
+              "description": "Vertex y-coordinate.",
+              "type": "number"
+            }
+          },
+          "type": [
+            "object",
+            "null"
+          ]
+        }
+      },
+      "type": "object"
+    },
+    "success": {
+      "description": "Whether the calculation succeeded.",
+      "type": "boolean"
+    }
+  },
+  "type": "object"
+}
  2. First observedv0.1.0

TDQS

A4.5/5.0
Behavior4/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare readOnlyHint=true, destructiveHint=false, and idempotentHint=true. The description adds valuable behavioral context: 'Runs locally on the numbers you provide: read-only, non-destructive, contacts no external service, and is rate-limited (60 requests/minute).' It also details the output for the solve operation, which is not in annotations.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is concise at 8 sentences, front-loaded with purpose, structured by operation, and every sentence adds value. It covers purpose, operations, outputs, usage guidance, and behavioral traits without redundancy.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness5/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

The tool has three operations and 7 parameters. The description explains all operations, required parameters per operation, degenerate cases (a=0), output details, and behavioral traits. It is completely adequate given the schema and annotations.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 100%, so the baseline is 3. The description adds high-level context about which operation uses which parameters but does not deepen the meaning of individual parameters beyond what the schema already provides.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description clearly states it solves quadratic equations, evaluates f(x), and builds monic quadratics from roots. It explicitly distinguishes from sibling tools math_scientific_calculator and math_matrix_calculator, which prevents confusion.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines5/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

The description provides explicit when-to-use instructions: 'Use this for second-degree polynomial algebra; use math_scientific_calculator for general expression evaluation and math_matrix_calculator for linear algebra.' It also explains the three operations and their required parameters.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.

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