Quadratic Formula Calculator
The quadratic formula solves any ax² + bx + c = 0: x² − 5x + 6 = 0 has roots x = 2, x = 3.
x = 2, x = 3
- Number of real roots
- 2
- Discriminant b² − 4ac
- 1
- Vertex (h, k)
- (5/2, -1/4)
- Axis of symmetry
- x = 5/2
- Sum of roots −b/a
- 5
- Product of roots c/a
- 6
- Factored form
- (x − 2)(x − 3)
Step-by-step solution
- x = [−b ± √(b² − 4ac)] ÷ 2a
- b² − 4ac = (-5)² − 4(1)(6) = 1
- x = [5 ± √1] ÷ 2 → x = 2, x = 3
Quadratic Formula Formula
x = [−b ± √(b² − 4ac)] ÷ 2a
discriminant b² − 4ac: > 0 two real roots, = 0 one, < 0 complex
Worked Example
The equation x² − 5x + 6 = 0:
- x = [−b ± √(b² − 4ac)] ÷ 2a
- b² − 4ac = (-5)² − 4(1)(6) = 1
- x = [5 ± √1] ÷ 2 → x = 2, x = 3
Discriminant b² − 4ac: 1 · Vertex (h, k): (5/2, -1/4) · Axis of symmetry: x = 5/2 · Sum of roots −b/a: 5 · Product of roots c/a: 6 · Factored form: (x − 2)(x − 3)
Test-day tip
On the digital SAT, graph y = ax² + bx + c and click where it crosses the x-axis to read the roots, or click the turning point for the vertex. For “how many solutions” questions, the discriminant is faster than solving.
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Example: The square root of 72: Simplified = 6√2.
Frequently asked questions
What does the discriminant tell you?
b² − 4ac > 0 gives two real roots, = 0 one repeated root (the vertex touches the x-axis), and < 0 no real roots (two complex roots).
How do I find the sum and product of the roots quickly?
For ax² + bx + c = 0 the roots add to −b/a and multiply to c/a, without solving the equation.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Quadratic Formula Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/quadratic-formula-calculator/
IEEE
Scientific Converters, "Quadratic Formula Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/quadratic-formula-calculator/