How to generate a quadratic equation with rational solutions

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

What you'll achieve: 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 LearningLemur:

Before you begin

Requirements

• Basic knowledge of how to create a LearningLemur 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 LearningLemur
• Common Patterns and Best Practices
• Glossary of Commands