Math & Statistics Solvers with Steps

Derived by completing the square on the general quadratic equation ax² + bx + c = 0. It provides the exact algebraic solutions for all degree-2 polynomials.

A cornerstone of Euclidean geometry stating that the area of the square whose side is the hypotenuse equals the sum of the areas of the squares on the other two legs.

Relates the ratios of the lengths of sides to the sines of their opposite angles in any planar triangle. Useful for surveying and navigation.

A general generalization of the Pythagorean theorem applicable to any planar triangle. When angle C = 90°, cos(90°) = 0, reducing exactly to a² + b² = c².

Derived directly from the Pythagorean theorem by treating the horizontal difference (Δx) and vertical difference (Δy) as perpendicular triangle legs.

Calculates the arithmetic mean of the x-coordinates and the y-coordinates respectively to find the center of mass or midpoint of a segment.

Measures the vertical change divided by the horizontal change ("rise over run"). A vertical line has an undefined slope where x₁ = x₂.

Proven through integration and exhaustion methods, showing that circular area equals half the circumference times the radius: (2πr · r) / 2 = πr².

Any triangle can be duplicated and reflected to form a parallelogram of base b and height h. Since parallelogram area is b · h, the triangle is exactly half.

Models linear non-compounding interest. Total future balance is calculated as A = P(1 + rt).

Captures geometric exponential growth over discrete intervals. As n approaches infinity, it converges to continuous compounding A = P · e^(rt).

Uses Bessel’s correction (n - 1 in the denominator) to provide an unbiased estimator of the true population variance from a subset sample.

The mathematical center of mass of a dataset. Sensitive to extreme skewed values or outliers, where median is often preferred.

Counts ordered sequences without replacement from a finite set of n distinct elements.

Divides the permutation count by r! to eliminate duplicate arrangements of the same subset of items.

Since a^m represents m factors of a and a^n represents n factors of a, multiplying them gives (m + n) total factors of a.

Logarithms are the inverses of exponential operations. The logarithm of a product equals the sum of the individual logarithms.