Quadratic formula calculator
Roots of ax² + bx + c = 0 with the quadratic formula, plus discriminant and vertex.
Two real roots
The equation x² − 5x + 6 = 0 has a positive discriminant and two real roots: x₁ = 3 and x₂ = 2.
How this is calculated →- Δ = b² − 4ac1
- Vertex (h, k)(2.5; −0.25)
- Sum of roots (−b/a)5
- Product of roots (c/a)6
The quadratic formula
Work out the discriminant Δ = b² − 4ac, then x = (−b ± √Δ) ÷ 2a. For x² − 5x + 6 = 0: Δ = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, that is, 3 and 2.
If Δ > 0 there are two real roots; if Δ = 0, one double root; if Δ < 0, no real roots (two complex conjugates). The parabola's vertex is at (−b/2a, −Δ/4a).
Frequently asked questions
What is the discriminant?
It is b² − 4ac. Its sign tells how many real roots the equation has.
What if b or c is zero?
The formula works the same. With c = 0, one root is zero; with b = 0, the roots are opposites.
What is a complex root?
When Δ < 0, the parabola does not cross the x-axis. The solutions involve i, the imaginary unit (i² = −1).
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