How to generate a quadratic equation with rational solutions

Use this method when you want to generate quadratic equations whose solutions are guaranteed to be rational numbers (instead of irrational or complex roots).

Following this guide, you will create a quadratic equation $ax^2 + bx + c = 0$where the discriminant is always a perfect square, ensuring that the solutions are rational.

See it in action: Watch how this logic is implemented inside Nubric:

Before you begin

Requirements

• Basic knowledge of how to create a Nubric question.
• Familiarity with adding an algorithm to generate random variables.

Steps

Generate random coefficients.

a = random([-10..10]/[0])
b = random([-10..10]/[0])
c = random([-10..10]/[0])

This generates non-zero coefficients for the quadratic equation.

Enforce a perfect square discriminant.

while square?(b^2-4*a*c) == false
do c = random([-10..10]/[0])
end

Compute the solutions.

sol1 = (-b + sqrt(b^2-4*a*c)) / (2*a)
sol2 = (-b - sqrt(b^2-4*a*c)) / (2*a)

These values can be used in grading logic or as expected answers.

Verify it worked

• Preview the question multiple times.
• Confirm that the discriminant is always a perfect square.
• Check that the computed solutions are rational numbers.
• Ensure no irrational square roots appear in previews.

Full algorithm (copy-paste version)

Use the complete version below if you want to copy the logic directly into your question algorithm:

# Generate random values for the coefficients
a = random([-10..10]/[0])
b = random([-10..10]/[0])
c = random([-10..10]/[0])

# Ensure the discriminant is a perfect square
while square?(b^2-4*a*c) == false
do c = random([-10..10]/[0])
end

# Compute solutions
sol1 = (-b + sqrt(b^2-4*a*c)) / (2*a)
sol2 = (-b - sqrt(b^2-4*a*c)) / (2*a)

Options and variations

• If you want integer solutions only, you can add additional constraints to ensure the solutions evaluate to integers (e.g., restrict coefficient intervals further).
• If you want only one repeated root, you can force the discriminant to equal 0 instead of being a perfect square.
• If you want smaller coefficients, you can reduce the interval (e.g., [-5..5]/[0]) to simplify calculations.

Common errors

Infinite loop in the while statement.
The interval may be too restrictive → Broaden the coefficient range

Unexpected irrational solutions.
Check that square?(...) is correctly written and applied to the discriminant

Division by zero error.
Ensure a is never 0 (already handled by [0] exclusion)

Related

• Understanding Advanced Logic in Nubric
• Common Patterns and Best Practices
• Glossary of Commands