Reference

Step-by-step references across 13 categories.

Algebra: Equations

Linear Equations

Solve equations of the form ax + b = c by isolating the variable.

x+1=5
2x+3=7

Quadratic Equations

Solve ax^2 + bx + c = 0 using the quadratic formula.

x^2+5x+6=0
x^2-2=0

Algebra: Logarithms

Evaluate Logarithms

Compute log values like log(100, 10) = 2.

log(100, 10)
log(E)

Expand Logarithms

Apply product, quotient, and power rules to expand log expressions.

log(x*y)
log(x/y)

Algebra: Exponents

Simplify Exponents

Combine exponent rules: product, quotient, power of a power.

x^2 * x^3
x^5 / x^2

Algebra: Imaginary Numbers

Simplify Imaginary Numbers

Apply i^2 = -1 to reduce powers of i.

I^3
I^5

Algebra: Polynomials

Expand Polynomials

Distribute and combine like terms.

expand((x+1)^3)
expand((x-2)(x+3))

Factor Polynomials

Find factors including difference-of-squares and trinomials.

factor(x^2-1)
factor(x^2+5x+6)

Algebra: Trigonometry

Evaluate Trig Functions

Compute exact values at standard angles.

sin(pi/4)
cos(pi)

Simplify Trig Expressions

Apply Pythagorean and other identities.

sin(x)^2 + cos(x)^2
tan(x) * cos(x)

Calculus

Differentiation

Apply power, product, quotient, and chain rules.

diff(x^3, x)
diff(x^2*sin(x), x)

Integration

Compute indefinite integrals.

integrate(x^2, x)
int(cos(x), x)

Limits

Evaluate limits at a point or at infinity, with a direct-substitution check first.

limit(sin(x)/x, x, 0)
limit(1/x, x, oo)

Linear Algebra: Matrices & Systems

Matrix Determinant

Compute 2x2 and 3x3 determinants via the ad - bc rule or cofactor expansion.

det([[1,2],[3,4]])
det([[2,0],[1,3]])

Matrix Inverse

Invert a square matrix with the adjugate formula, or detect that it is singular.

inv([[1,2],[3,4]])
inv([[4,7],[2,6]])

Matrix Transpose

Swap rows and columns of a matrix.

transpose([[1,2],[3,4]])
transpose([[1,2,3],[4,5,6]])

Matrix Multiplication

Multiply two matrices row by column, entry by entry.

[[1,2],[3,4]] * [[5,6],[7,8]]
[[1,0],[0,1]] * [[2,3],[4,5]]

Systems of Linear Equations

Solve systems of 2 or 3 linear equations by Gaussian elimination. Separate equations with a semicolon.

x + y = 3; x - y = 1
2x + y = 1; x - y = 1

Eigenvalues & Eigenvectors

Characteristic polynomial, eigenvalues and eigenvectors of 2x2 and 3x3 matrices.

eigenvalues([[2,1],[1,2]])
eigenvectors([[2,1],[1,2]])

Calculus: Differential Equations

Separable Differential Equations

Solve first-order separable ODEs by separating variables and integrating. Use y' for the derivative.

y' = x*y
y' = 2*y

First-Order Linear Differential Equations

Solve y' + P(x)y = Q(x) with an integrating factor.

y' + 2*y = x
y' + y = x

Second-Order Linear ODEs

Constant-coefficient homogeneous equations via the characteristic equation.

y'' + 3*y' + 2*y = 0
y'' + 2*y' + y = 0

Probability & Statistics

Descriptive Statistics

Mean, median, mode, variance and standard deviation of a data set.

stats(2, 4, 4, 5, 7)
mean(1, 2, 3, 4)

Combinations & Permutations

Factorials, combinations nCr and permutations nPr with expanded factorial steps.

nCr(5, 2)
nPr(5, 2)

Binomial Probability

P(X = k) for n independent trials with success probability p.

binomial_p(10, 3, 1/2)
binomial_p(6, 2, 1/3)

Geometry

Right Triangles

Find the hypotenuse or a leg with the Pythagorean theorem.

hypotenuse(3, 4)
leg(5, 3)

Triangle Area (Heron)

Area of a triangle from its three sides using Heron's formula.

triangle_area(3, 4, 5)
triangle_area(7, 8, 9)

Circles

Area and circumference of a circle, exact (in terms of pi) and approximate.

circle_area(3)
circle_circumference(3)

Coordinate Geometry

Distance, midpoint and slope between two points.

distance(1, 2, 4, 6)
midpoint(1, 2, 3, 4)

Algebra: Inequalities

Inequalities

Solve linear and quadratic inequalities; answers as intervals.

2x + 3 < 11
2x + 3 <= 11

Algebra: Sequences & Series

Sequences & Series

nth term and partial sum of arithmetic and geometric sequences.

arithmetic_seq(2, 5, 10)
geometric_seq(3, 2, 8)