What is a Quadratic Equation Calculator?
This calculator solves ax² + bx + c = 0 and shows the discriminant.
How it's calculated
Quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a
Discriminant D = b² − 4ac
Step-by-step example
Solving x² − 5x + 6 = 0
- Step 1 Discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1
- Step 2 Since D>0, two distinct real roots: x = (5 ± 1) ÷ 2
- Step 3 Solve: x = 6÷2 = 3 or x = 4÷2 = 2
Interpretation: Checking (x−2)(x−3)=x²−5x+6 confirms this matches the original equation.
The discriminant tells you everything at a glance
Without solving for the roots directly, the discriminant alone tells you how many solutions exist and what type. D > 0 means two distinct real roots, D = 0 means one repeated root, and D < 0 means no real roots (two complex roots instead). When the value under the square root is negative, there's no real-number answer — this is exactly where imaginary numbers (i) enter the picture.
Good to know
- The quadratic formula is simply the result of completing the square. It's not something to memorize blindly — it's a derived conclusion you can reconstruct from first principles.
- Babylonian clay tablets from around 2000 BCE already show methods for solving quadratic equations, though they only handled positive solutions since negative numbers weren't yet conceptualized.
- Vieta's formulas let you find properties of the roots without solving explicitly: the sum of the roots equals −b/a, and their product equals c/a — exactly what you're doing when you factor by 'finding two numbers that add to b and multiply to c.'
- Cubic and quartic equations also have general formulas, but they're extremely complex — and Galois and Abel proved that no general formula exists for degree 5 and higher, one of mathematics' most striking impossibility results.
- Real-world problems involving maximum or minimum values — projectile trajectories, optimal pricing, braking distance — very often reduce to quadratic equations.
Frequently asked questions
Q. Does a negative discriminant mean there's no answer?
No real solutions exist, but two complex (imaginary) solutions do.
Q. Is there a way to solve without the quadratic formula?
If the expression factors easily, factoring is much faster. The formula is the fallback when it doesn't.
Q. What happens if a = 0?
The equation loses its quadratic term and becomes linear. A true quadratic equation requires a ≠ 0.