Math & Geometry
Essential math solvers for students, engineers, and everyday problem-solving.
Solve ax² + bx + c = 0
Max 50 terms.
Quadratic Formula Explained
The quadratic formula x = (−b ± √(b²−4ac)) / 2a solves any equation in the form ax² + bx + c = 0. The discriminant (b²−4ac) tells you the nature of roots: positive → two real roots; zero → one repeated real root; negative → two complex (imaginary) roots. The vertex of the parabola is at x = −b/2a, y = c − b²/4a.
Heron's Formula for Triangle Area
When you know all three side lengths of a triangle, Heron's formula gives the area without needing height: s = (a+b+c)/2 (semi-perimeter), Area = √(s(s−a)(s−b)(s−c)). Triangle types: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (all different). The triangle inequality requires each side to be less than the sum of the other two.
Circle, Sphere & Cylinder Formulas
Circle: Area = πr², Circumference = 2πr. Sphere: Surface Area = 4πr², Volume = (4/3)πr³. Cylinder: Surface Area = 2πr² + 2πrh, Volume = πr²h. Note: π ≈ 3.14159265. All measurements are in the square or cubic version of the input unit.
Percentage Change vs. Percentage Difference
Percentage Change = ((New − Old) / |Old|) × 100. A positive result means increase; negative means decrease. This is different from Percentage Difference, which is used when neither value is the "reference" and is calculated as |V1−V2| / ((V1+V2)/2) × 100. Reverse percentage is useful for finding the original price before a discount or markup was applied.
Fibonacci & The Golden Ratio
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) is defined by F(n) = F(n−1) + F(n−2). As the sequence grows, the ratio of consecutive terms converges to the Golden Ratio φ ≈ 1.6180339887. This ratio appears throughout nature (phyllotaxis, shell spirals, tree branching), art, architecture, and financial technical analysis (Fibonacci retracement levels).
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