Quadratic equation (Bhaskara) calculator
Solve quadratic equations (ax² + bx + c = 0) with the delta and roots step by step.
How the calculation works
Bhaskara's formula (the quadratic formula) solves any second-degree equation in the form ax² + bx + c = 0 (with a ≠ 0). It has two steps.
1) Discriminant (delta):
Δ = b² − 4ac
2) Roots:
x = (−b ± √Δ) / 2a
The ± sign yields the two roots.
What delta tells you
- Δ > 0: two distinct real roots
- Δ = 0: one (double) real root
- Δ < 0: no real roots (they are complex)
Step by step: x² − 5x + 6 = 0
Here a = 1, b = −5, c = 6:
- Δ = (−5)² − 4·1·6 = 25 − 24 = 1
- √Δ = 1
- x = (5 ± 1) ÷ 2 → x₁ = 3, x₂ = 2
Examples
- x² − 5x + 6 = 0: Δ = 1 → roots 3 and 2.
- x² − 4x + 4 = 0: Δ = 0 → double root x = 2.
- x² + x + 1 = 0: Δ = −3 → no real roots.
Frequently asked questions
What is the delta?
It is the discriminant (b² − 4ac). Its sign tells how many real roots the equation has.
How do I set up the equation from a word problem?
Move everything to one side, equal to zero, and read the coefficients: a multiplies x², b multiplies x and c is the constant.
Does every quadratic need Bhaskara?
No. When c = 0 or b = 0 you can factor or take a direct root, which is faster. Bhaskara works for any case.
What if delta is negative?
There is no solution in the real numbers. The roots exist only in the complex numbers.
What does a negative discriminant mean?
The equation has no real roots — the parabola never crosses the x-axis. The roots exist in the complex numbers.
When should I use sum and product instead?
When the roots look like simple integers. Find two numbers summing to −b/a and multiplying to c/a.
How do I find the vertex?
At x = −b/2a, midway between the roots. It is the minimum when a is positive and the maximum when negative.
What is the most common mistake?
Forgetting the sign of b. In x² − 5x + 6, b is −5, so −b becomes +5.
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Updated on June 17, 2026 · by Rafael Rossi · Methodology & sources