Quadratic Formula — ax² + bx + c = 0
x = (−b ± √(b²−4ac)) / 2a
Variable Key
a= coefficient of x²b= coefficient of xc= constantΔ= discriminant = b² − 4ac
Quadratic formula
Finds both roots simultaneously.
Discriminant analysis
Determines the nature of roots before solving.
Δ > 0
Δ = 0
Δ < 0
Vieta's formulas
Relationships between roots and coefficients.
Sum of roots
Product of roots
Vertex form
Rewrite in vertex form to find the parabola's turning point.
Vertex
a= coefficient of x²
b= coefficient of x
c= constant
Δ= discriminant = b² − 4ac
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x. It works for all quadratics — even ones that cannot be factored — making it the most universal solving method. The formula was known to Babylonian mathematicians as early as 2000 BC.
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Fun Fact
The quadratic formula was first written in modern algebraic notation by René Descartes in 1637. Before that, mathematicians described the same method in words and geometric diagrams.
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