Algebra Studio – a complete algebra calculator

Operators
+ - * / ^ and brackets. Implicit multiplication works: 3x, 2(x+1), x(x-2), 2xy.

Functions
sqrt cbrt abs exp ln log log10 log2 sin cos tan asin acos atan sinh cosh tanh floor ceil round sign min max pow mod
log(x) is base 10, log(x, b) any base, ln natural. Trig uses radians.

Constants & numbers
pi, e, fractions like 3/4, powers like 1.5e3. Number boxes accept expressions such as pi/2 or 2^10.
Nothing is ever passed to eval.

Key formulas

Distributive law: a(b + c) = ab + ac
(a + b)² = a² + 2ab + b²
(a − b)(a + b) = a² − b²
Like terms combine: 3x + 5x = 8x

Evaluate an expression

Substitute values for variables and see every arithmetic step.

Expand & simplify

Multiply out brackets, combine like terms (any number of variables), list the terms and the common factor.

Are two expressions equivalent?

Exact check for polynomials, plus random-point testing for everything else.

Key formulas

ax + b = 0 ⇒ x = −b/a
Slope-intercept: y = mx + b
Cramer (2×2): x = Dₓ/D, y = D_y/D
System has a unique solution when det ≠ 0

Solve a linear equation

One unknown, brackets and variables on both sides allowed.

Solve a system of linear equations

Two to six unknowns (any letters), one equation per line. Uses Gauss–Jordan elimination and detects unique / none / infinite solutions.

Line from slope & intercept

Table of values, intercepts and graph for y = mx + b.

Key formulas

x = (−b ± √(b² − 4ac)) / 2a
Discriminant Δ = b² − 4ac
Vertex: (−b/2a, c − b²/4a)
Vieta: x₁ + x₂ = −b/a, x₁x₂ = c/a

Quadratic ax² + bx + c = 0

Roots (real or complex), discriminant, vertex, forms and graph.

Polynomial roots (degree up to 10)

Type any polynomial equation; finds all real and complex roots with multiplicities (Durand–Kerner + clustering).

Build a quadratic from its roots

Given the roots and a leading coefficient, get the expanded form.

Key formulas

Function: each x has exactly one f(x)
Even: f(−x) = f(x) Odd: f(−x) = −f(x)
f′(x) ≈ [f(x+h) − f(x−h)] / 2h
Zeros: solutions of f(x) = 0

Analyse f(x) on an interval

Table, graph, zeros, turning points, increasing/decreasing intervals, parity and undefined points.

Value, derivative & integral at a point

Numerical probes: f(a), slope and tangent line, curvature, and the area under the curve (Simpson’s rule).

Solve f(x) = c numerically

Finds every solution in an interval – works for any continuous function, not only polynomials.

Key formulas

(f∘g)(x) = f(g(x))
In general f∘g ≠ g∘f
f and g are inverses ⇔ f(g(x)) = g(f(x)) = x
Domain of f∘g: x in dom g with g(x) in dom f

Compose f and g

Symbolic and numeric f(g(x)) and g(f(x)), with graphs.

Triple composition f(g(h(x)))

Chain three functions and expand the result when it is a polynomial.

Are f and g inverses?

Tests f(g(x)) = x and g(f(x)) = x, and gives the inverse of a linear f.

Key formulas

y = a·bˣ (b > 1 growth, 0 < b < 1 decay)
A = P(1 + r/n)^(nt) A = Pe^(rt)
Doubling time = ln 2 / ln b
Through (x₁,y₁),(x₂,y₂): b = (y₂/y₁)^(1/(x₂−x₁))

Evaluate y = a·bˣ

Growth/decay analysis, doubling or half-life and graph.

Solve a·bˣ = y for x

Uses logarithms: x = ln(y/a) / ln b.

Compound interest & growth

Periodic or continuous compounding, effective rate, doubling time, year-by-year balance.

Fit y = a·bˣ through two points

Find the exponential model and predict any value.

Key formulas

log_b x = y ⇔ bʸ = x
log(MN) = log M + log N
log(M/N) = log M − log N, log(Mᵖ) = p·log M
Change of base: log_b x = ln x / ln b

Logarithm calculator

Any base, with the change-of-base working shown.

Solve log_b(ax + c) = y

Converts to exponential form and checks the domain.

Solve b^(ax + c) = y

Take logs of both sides.

Verify the log laws numerically

Both sides of each law, side by side.

Graph log and its inverse

y = log_b x and y = bˣ mirror each other in y = x.

Logarithms in science

pH, decibels and earthquake magnitude.

Key formulas

Arithmetic: aₙ = a₁ + (n−1)d, Sₙ = n(a₁ + aₙ)/2
Geometric: aₙ = a₁rⁿ⁻¹, Sₙ = a₁(1 − rⁿ)/(1 − r)
Infinite geometric (|r|<1): S = a₁/(1 − r)
Σᵢ₌₁ⁿ i = n(n+1)/2, Σ i² = n(n+1)(2n+1)/6

Arithmetic sequence & series

nth term, partial sum, and which term equals a target.

Geometric sequence & series

nth term, partial sum and the infinite sum when it converges.

Identify a sequence & predict the next terms

Detects arithmetic, geometric or polynomial patterns using finite differences and builds the formula aₙ.

Sigma notation: Σ f(i)

Sum (and product) of any expression in i over a range.

Recursive sequences

Iterate u = previous term, v = the term before it. Try "u+v" for Fibonacci or "u/2 + 3/u" for √6.

Key formulas

Multiplying/dividing by a negative flips the sign
|u| < c ⇔ −c < u < c
|u| > c ⇔ u < −c or u > c
Polynomial sign changes only at real roots

Solve an inequality (polynomial or chain)

Handles linear, quadratic and higher-degree polynomials, and chains like −3 < 2x + 1 ≤ 7. Shows the sign chart and number line.

Absolute-value inequality |u| ⋚ c

u can be any polynomial in x. Shows the distance interpretation.

Systems of inequalities & linear programming

Enter constraints in x and y (one per line). The tool finds the corner points of the feasible region and optimises a linear objective.

Key formulas

d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope m = (y₂−y₁)/(x₂−x₁)
Circle: (x−h)² + (y−k)² = r²
Point–line distance = |ax₀+by₀+c| / √(a²+b²)

Two points: distance, midpoint, line

Everything about the segment and the line through two points, plus a point dividing it in a ratio.

Triangle from three vertices

Sides, area (shoelace), angles, classification, centroid, circumcircle and incircle.

Circle from centre & radius (or general form)

Equation forms, area, intercepts and where a test point lies. Leave D, E, F blank to use centre & radius.

Line meets circle

Secant, tangent or miss – with the intersection points and chord length.

Point and line: distance, foot, reflection

Perpendicular distance from a point to ax + by + c = 0.

Transform a shape

Translate, scale, rotate or reflect a list of points about any centre.

Algebra Studio · PHP + Bootstrap 5 · all calculations run on the server with a safe expression parser.