Random Quadratic Equation Exercise Generator

Construct monic quadratics from bounded integer roots and compute verified solution records.

Description

Build monic quadratic equations from bounded integer roots and return their computed solutions.

A quadratic equation ax² + bx + c = 0 can be constructed from its roots. Expanding (x − r1)(x − r2) gives a monic equation, meaning a = 1, with b determined by the root sum and c by their product.^[1]

This generator samples integer roots first and then constructs the coefficients. Each item in result.exercises has coefficients and solution, including case, discriminant, and roots. It produces real integer-root questions, including repeated roots, rather than arbitrary coefficient triples.

When to use Random Quadratic Equation Exercise Generator

Use it for factoring or quadratic-solution worksheets with an included answer key and controlled root magnitude.

How to use Random Quadratic Equation Exercise Generator and understand its results

Enter integer exerciseCount from 1 through 64 and integer rootMagnitude from 1 through 1,000.
Run and read coefficients.a, coefficients.b, and coefficients.c as ax² + bx + c = 0.
Keep solution separate for checking. Read roots and case; a double_root result has one root entry even though its multiplicity is two.

r1,r2 = −M + floor(randomDraw × (2M + 1)); a = 1; b = −(r1 + r2); c = r1 × r2

Two independent draws choose roots in inclusive [−M, M].

The quadratic-equation calculator computes the solution from those coefficients. The discriminant is (r1 − r2)², so the construction yields either two real roots or a double root, never a nonreal pair. Roots are returned in ascending order when distinct.

json

{
  "input": {
    "exerciseCount": 3,
    "rootMagnitude": 10
  },
  "randomDraw": 0.25,
  "output": {
    "result": {
      "exercises": [
        {
          "coefficients": {
            "a": 1,
            "b": 10,
            "c": 25
          },
          "solution": {
            "case": "double_root",
            "discriminant": 0,
            "imaginaryPart": 0,
            "realPart": 0,
            "roots": [
              -5
            ]
          }
        },
        {
          "coefficients": {
            "a": 1,
            "b": 10,
            "c": 25
          },
          "solution": {
            "case": "double_root",
            "discriminant": 0,
            "imaginaryPart": 0,
            "realPart": 0,
            "roots": [
              -5
            ]
          }
        },
        {
          "coefficients": {
            "a": 1,
            "b": 10,
            "c": 25
          },
          "solution": {
            "case": "double_root",
            "discriminant": 0,
            "imaginaryPart": 0,
            "realPart": 0,
            "roots": [
              -5
            ]
          }
        }
      ]
    }
  }
}

Zero and repeated roots are allowed. rootMagnitude bounds the sampled roots: |b| can reach 2M and |c| can reach M². Exchanging two sampled roots produces the same equation, so equations are not uniformly distributed over coefficient triples.

Limitations and assumptions

The default source is Math.random; no seed input or cryptographic entropy guarantee is provided. Repeated runs may select the same result, and the distribution depends on the random source.

Only monic integer-root equations are generated. There are no complex-root, nonmonic, symbolic, automatically hidden-answer, or difficulty-scored exercises. The realPart and imaginaryPart placeholders are not substitutes for the returned real roots.

Alternative or Complementary methods

Solve a chosen equation with the deterministic calculator. Polynomial-evaluation questions support a separate substitution exercise rather than solving for roots.

References

Algebra and Trigonometry 2e: Quadratic Functions — OpenStax, Rice University

Similar or alternative tools

Quadratic Equation
Solve explicit quadratic coefficients.
Random Polynomial Evaluation Exercise Generator
Generate bounded polynomial substitution questions.