LifeTally
Real roots of a quadratic equation
Which real x satisfy your equation ax² + bx + c = 0?
Enter three coefficients. Find the discriminant, the number of distinct real roots and their ordered values.
Your coefficients
Write the equation in the form ax² + bx + c = 0 before entering its coefficients. Keep their signs. Use consistent quantities in that equation; no units are converted. Enter fractions as decimals, not a/b.
How the estimate works
D = b² − 4ac. For a ≠ 0: D < 0 gives no real roots; D = 0 gives the repeated root −b/(2a); D > 0 gives (−b ± √D)/(2a). Results are ordered from smaller to larger, also for negative a. All three coefficients are necessary. Classification uses the unrounded discriminant.
ax² + bx + c = 0; a ≠ 0
D = b² − 4ac
x = (−b ± √D) ÷ (2a); D ≥ 0Only a scalar real quadratic equation. No linear fallback, complex-root output, symbolic factoring or derivation, graph, curve fitting, coefficient inference, uncertainty estimate, measurement validation, physical prediction or safety advice. Entering a mathematical equation does not validate a real-world model.
OpenStax: quadratic formula ↗OpenStax: discriminant and real-root count ↗
Check the filled example
Our chosen example: a = 2, b = −5, c = −3 gives 2x² − 5x − 3 = 0. D = 25 + 24 = 49; the real roots are −0.5 and 3. Substitution gives 0 for each. These are chosen mathematical coefficients, not a norm or physical forecast.
Accepted input values
- Coefficient a
- 2
- Coefficient b
- -5
- Constant c
- -3
Plan the next step
Use the standard calculator for separate arithmetic. Check that your original equation has been moved to the stated form.
Open the standard calculator →