Science and math questions, solved step by step.
Worked solutions in mathematics, statistics, physics, chemistry, biology, computer science and LaTeX, each checked before publication, plus free calculators that explain every step.
Browse by topic
All topics →Mathematics
Algebra, calculus, analysis, topology, linear algebra and number theory, worked out step by step.
Statistics & Probability
Distributions, estimators, expected values and hypothesis tests with clear derivations.
Physics
Mechanics, electromagnetism, thermodynamics, relativity and quantum physics explained.
Chemistry
Reactions, bonding, equilibrium, thermochemistry and lab calculations.
Biology
Evolution, biochemistry, physiology and ecology, answered with evidence.
Computer Science
Algorithms, complexity, automata, number representation and digital logic.
LaTeX & TikZ
Working solutions for tables, figures, bibliographies, TikZ and pgfplots.
Academia
Publishing, peer review, PhD life and research careers.
Calculators
All calculators →Quartiles
Find Q1, median, Q3 and IQR from a list of values or from histogram bins (grouped data), with every step shown.
Standard deviation
Sample and population standard deviation, variance and mean, with the full sum-of-squares working.
Normal probability
Given the mean and standard deviation, find P(X < b), P(X > a) or P(a < X < b) with z-scores.
Dice probability
Exact distribution, mean and standard deviation for the sum of any number of fair dice.
Matrix
Determinant with cofactor expansion, inverse, symmetric eigen-decomposition and singular values.
Boolean simplifier
Turn minterms or a truth table into a minimal sum-of-products expression using Quine–McCluskey.
Recently updated
- How to Prove a Homogeneous System Has Infinitely Many SolutionsFor a homogeneous system with n unknowns and m equations over an infinite field, rank-nullity guarantees a nonzero solution and infinitely many scalar multiples.
- When Is $b^2c^2+c^2a^2+a^2b^2>abc(a+b+c)$ True?A sum-of-squares identity proves the expression is at least the right-hand side; equality occurs at a=b=c, so strictness needs an exception.
- How to Go from a PDF to Its Source in TeXstudioUse SyncTeX in TeXstudio to connect a PDF location to its LaTeX source; enable it, recompile, then Ctrl-click the PDF text.
- How to Rerun Biber on a LaTeX FileTo rerun Biber, compile your LaTeX document once to create its .bcf file, run Biber on the document basename, then compile LaTeX again.
- How to Reduce Space Between a Proof and an Enumerated ListUse enumitem’s topsep option to reduce space around an enumerate list; adjust the proof environment too if its bottom spacing remains.
- How to Show That an Integral-Defined Function Is Strictly DecreasingFor a function defined by an integral, prove strict decrease by comparing its integrands or justifying differentiation under the integral sign.
- How to Solve $8n^2=64n\log_2(n)$Divide by 8n to reduce the equation to n = 8 log base 2 of n; it has two positive real solutions, approximately 1.1000 and 43.5593.
- How to Solve an Equation with Exponential and Polynomial TermsLearn when mixed exponential-polynomial equations have exact solutions, how to locate all real roots, and how to approximate them reliably.
- How to Solve Integrals Using SeriesUse a power series to replace a difficult integrand with simpler terms, then integrate term by term after checking convergence on the interval.
- How to Write a Cover Letter for a Canadian Study PermitWrite a concise Canadian study permit cover letter that explains your program choice, funding, and application details without replacing required documents.
- How to Write Equations for Metals Reacting With WaterLearn how to write and balance equations for metals reacting with cold water, hot water, or steam, with examples and common exceptions.
- How to Write Pseudocode in LaTeX with TexmakerWrite pseudocode in Texmaker by loading the algorithm and algpseudocode packages, then using their environments and commands in a LaTeX document.