Solve ax² + bx + c = 0 using the quadratic formula
a =
b =
c =
Result
Equation
Roots
Step-by-Step Derivation
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
The quadratic formula solves any equation in the form ax²+bx+c=0 where a is not zero. The expression b²-4ac is called the discriminant, and it controls the type of roots before the final division is done. A positive discriminant gives two real solutions, a zero discriminant gives one repeated real solution, and a negative discriminant gives two complex solutions. This makes the formula a reliable method even when factoring is difficult.
⚠Note: a must not be 0. If a=0, the expression is linear rather than quadratic.
What This Calculator Shows
Discriminant
Computes Δ=b²-4ac and explains the root type.
Real or Complex Roots
Displays exact root structure and decimal values when appropriate.
Vertex
Finds the x-coordinate -b/(2a), useful for graphing parabolas.
Vieta Check
Verifies that root sum equals -b/a and product equals c/a.
💡 Example: x²-5x+6=0 has Δ=1, so x=(5±1)/2. The roots are x=3 and x=2.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.