Quadratic Formula Calculator
Solve quadratic equations ax²+bx+c=0 with step-by-step solution, discriminant, and roots
Enter Coefficients
ax² + bx + c = 0
Quick examples:
Solution
Discriminant (Δ)
1
Two distinct real roots
Sum of roots
5
= -b/a
Product of roots
6
= c/a
x₁
3
x₂
2
Vertex of parabola
(2.5, -0.25)
Axis of symmetry: x = 2.5
Step-by-step solution
- 1. Standard form: ax² + bx + c = 0
- 2. a=1, b=-5, c=6
- 3. Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
- 4. Δ > 0 → Two distinct real roots
- 5. x = (-b ± √Δ) / 2a = (5 ± √1) / 2
- 6. x₁ = 3, x₂ = 2
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b²−4ac)) / (2a). It gives the solutions to any equation of the form ax² + bx + c = 0, where a ≠ 0.
What does the discriminant tell you?
The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives two complex conjugate roots.
What if a = 0?
When a = 0 the equation reduces to bx + c = 0, which is linear (not quadratic). The calculator handles this case and solves for x = −c/b.
What are complex roots?
When the discriminant is negative, the square root involves imaginary numbers. The two roots are complex conjugates of the form p ± qi where i = √(−1).