# Math Solving Space

> Free online maths calculators and step-by-step solvers


## Posts

- [Random Numbers Explained: Fair Picks, Dice and Lotteries](https://mathsolving.space/blog/random-numbers-and-probability/): What makes numbers random, how to pick fair random numbers in a range, and how probability explains dice rolls and lottery odds.
- [How to Graph a Quadratic Function (Parabola) Step by Step](https://mathsolving.space/blog/how-to-graph-a-quadratic-function/): Graph y = ax² + bx + c by finding the vertex, axis of symmetry, intercepts and direction, with worked examples and an online quadratic grapher.
- [Vectors Explained: Magnitude, Dot Product and Cross Product](https://mathsolving.space/blog/vectors-dot-and-cross-product/): Learn what vectors are, how to find magnitude, the dot product and the angle between vectors, and how the 3D cross product works.
- [Taylor Series Explained: Approximating Functions with Polynomials](https://mathsolving.space/blog/taylor-series-explained/): What Taylor and Maclaurin series are, how to build them from derivatives, and why sin x ≈ x − x³/6 + x⁵/120 near zero. Clear examples.
- [Speed, Distance and Time: The Formula Triangle Explained](https://mathsolving.space/blog/speed-distance-time/): Use speed = distance ÷ time to find any of the three, convert units, work with hours and minutes, and read distance–time graphs.
- [How to Solve Linear Inequalities (and When to Flip the Sign)](https://mathsolving.space/blog/solving-linear-inequalities/): Solve linear inequalities step by step, learn the one rule that flips the sign, and show answers on a number line and in interval notation.
- [Scientific Notation (Standard Form) Explained](https://mathsolving.space/blog/scientific-notation-explained/): Write very large and very small numbers in scientific notation, convert back, and multiply and divide numbers in standard form.
- [Rounding and Significant Figures: Rules and Examples](https://mathsolving.space/blog/rounding-and-significant-figures/): How to round to decimal places, significant figures and the nearest 10, 100 or 1000, plus the rules for counting significant figures.
- [Roman Numerals Explained: How to Read and Write Them](https://mathsolving.space/blog/roman-numerals-explained/): Learn the seven Roman numeral symbols, the subtraction rule, and how to convert years like 2026 and 1994 to and from Roman numerals.
- [Riemann Sums Explained: Left, Right, Midpoint and Trapezoid](https://mathsolving.space/blog/riemann-sums-explained/): How Riemann sums approximate the area under a curve with rectangles, how left, right, midpoint and trapezoid rules compare, and why they lead to integrals.
- [Ratios and Proportions Made Simple](https://mathsolving.space/blog/ratios-and-proportions-made-simple/): Simplify ratios, share an amount in a ratio, and solve proportions by cross-multiplying, with recipe and map examples.
- [Probability Basics: And, Or, Not and Dice](https://mathsolving.space/blog/probability-basics-and-or-not/): Learn the basic probability rules for 'and', 'or' and 'not', independent and mutually exclusive events, and how to count dice outcomes.
- [The Poisson Distribution Explained with Examples](https://mathsolving.space/blog/poisson-distribution-explained/): When to use the Poisson distribution, how to calculate P(X = k) from an average rate λ, and how it compares with the binomial distribution.
- [Newton's Method Explained: Finding Roots with Tangent Lines](https://mathsolving.space/blog/newtons-method-explained/): How Newton's method finds roots of equations by following tangent lines, the formula, worked examples for √2 and cos x = x, and when it fails.
- [Line of Best Fit: Linear Regression and Correlation Explained](https://mathsolving.space/blog/line-of-best-fit-linear-regression/): How to find the least-squares line of best fit, what the correlation coefficient r and r² mean, and why correlation is not causation.
- [How to Solve Cubic Equations Using the Factor Theorem](https://mathsolving.space/blog/how-to-solve-cubic-equations/): Solve cubic equations by finding a root with the rational root theorem, dividing it out, and solving the remaining quadratic. Worked examples.
- [How to Find the Inverse of a Matrix (2×2 and 3×3)](https://mathsolving.space/blog/how-to-find-the-inverse-of-a-matrix/): Find the inverse of a 2×2 matrix with the quick formula and of a 3×3 matrix by Gauss–Jordan elimination, and check it with A × A⁻¹ = I.
- [How to Find the Determinant of a Matrix](https://mathsolving.space/blog/how-to-find-the-determinant-of-a-matrix/): Calculate 2×2 and 3×3 determinants with ad − bc and cofactor expansion, learn what a zero determinant means, and see it as an area.
- [How to Factor Polynomials: A Practical Checklist](https://mathsolving.space/blog/how-to-factor-polynomials/): A step-by-step checklist for factoring polynomials: common factors, differences of squares, quadratics, grouping and the factor theorem.
- [Graphing Sine and Cosine: Amplitude, Period and Phase Shift](https://mathsolving.space/blog/graphing-sine-and-cosine/): Learn how A, B, C and D change the graph of y = A sin(Bx + C) + D, with worked examples for amplitude, period, phase shift and midline.
- [Exponential Growth and Decay: Formula, Doubling Time and Half-Life](https://mathsolving.space/blog/exponential-growth-and-decay/): Understand exponential growth and decay, use A = A₀(1 + r)ᵗ, and find doubling times and half-lives with logarithms. Worked examples.
- [Percentages When Shopping: Discounts, Tax and Tips](https://mathsolving.space/blog/discounts-tax-and-tips/): Work out sale prices, stacked discounts, VAT or sales tax (adding and removing it) and tips quickly, with the multiplier method.
- [Confidence Intervals Explained (Without the Jargon)](https://mathsolving.space/blog/confidence-intervals-explained/): What a 95% confidence interval really means, how to calculate one from a sample mean, and how sample size changes the margin of error.
- [Complex Numbers for Beginners: i, Arithmetic and the Argand Diagram](https://mathsolving.space/blog/complex-numbers-for-beginners/): What i = √−1 means, how to add, multiply and divide complex numbers, and how modulus, argument and the Argand diagram work.
- [Completing the Square and Vertex Form, Step by Step](https://mathsolving.space/blog/completing-the-square-and-vertex-form/): Learn how to complete the square to write a quadratic in vertex form a(x − h)² + k, find the vertex, and see why it matters for graphs.
- [Box Plots and Quartiles: Five-Number Summary and Outliers](https://mathsolving.space/blog/box-plots-and-quartiles/): How to find quartiles and the interquartile range, draw a box-and-whisker plot, and spot outliers with the 1.5 × IQR rule.
- [The Binomial Distribution Explained Step by Step](https://mathsolving.space/blog/binomial-distribution-explained/): When to use the binomial distribution, the formula for exactly k successes, at-most and at-least probabilities, and the mean and standard deviation.
- [Binary and Hexadecimal Explained: Converting Number Bases](https://mathsolving.space/blog/binary-and-hexadecimal-explained/): How place value works in binary and hexadecimal, and how to convert between decimal, binary and hex step by step.
- [What Is the Discriminant? Reading a Quadratic at a Glance](https://mathsolving.space/blog/what-is-the-discriminant/): The discriminant b² − 4ac tells you how many roots a quadratic has and what kind they are. Learn to use it with clear examples.
- [Volume and Surface Area: The Formulas You Actually Need](https://mathsolving.space/blog/volume-and-surface-area-formulas/): Volume and surface area formulas for cubes, cuboids, cylinders, cones, spheres and pyramids, with worked examples and the one-third rule explained.
- [Three Ways to Solve a Quadratic Equation](https://mathsolving.space/blog/three-ways-to-solve-a-quadratic-equation/): Solve quadratic equations by factoring, the quadratic formula and completing the square, with worked examples and tips on choosing a method.
- [Standard Deviation and Variance Explained Step by Step](https://mathsolving.space/blog/standard-deviation-and-variance-explained/): What standard deviation measures, how to calculate it for a population or a sample, why samples divide by n − 1, and a full worked example.
- [Sine, Cosine and Tangent Explained (SOHCAHTOA and the Unit Circle)](https://mathsolving.space/blog/sine-cosine-tangent-explained/): Learn what sin, cos and tan mean in right triangles and on the unit circle, the exact values for 30°, 45° and 60°, and the ASTC sign rule.
- [The Pythagorean Theorem with Worked Examples](https://mathsolving.space/blog/pythagorean-theorem-with-worked-examples/): Use a² + b² = c² to find any side of a right triangle, check for right angles, and find distances on a grid. Worked examples and common mistakes.
- [Permutations vs Combinations: When Order Matters](https://mathsolving.space/blog/permutations-vs-combinations/): Learn the difference between permutations and combinations, the nPr and nCr formulas, and how to use them for lotteries, teams and passwords.
- [The Normal Distribution and Z-Scores Made Simple](https://mathsolving.space/blog/normal-distribution-and-z-scores/): Understand the bell curve, the 68–95–99.7 rule, z-scores and how to find normal probabilities, with worked examples on test scores and heights.
- [Logarithms Explained for Beginners](https://mathsolving.space/blog/logarithms-explained-for-beginners/): What a logarithm is, how it undoes an exponent, the three log laws, the change-of-base formula and how to solve exponential equations.
- [Limits Explained: Substitution, Factoring and L'Hôpital's Rule](https://mathsolving.space/blog/limits-and-lhopitals-rule/): What a limit is, how to evaluate one by substitution, factoring or L'Hôpital's rule, and how to handle limits at infinity. Clear worked examples.
- [Integration for Beginners: Antiderivatives and Area](https://mathsolving.space/blog/integration-for-beginners/): Understand integration as reverse differentiation and as area under a curve. Learn the power rule, definite integrals and integration by parts.
- [How to Solve Simultaneous Equations: Substitution vs Elimination](https://mathsolving.space/blog/how-to-solve-simultaneous-equations/): Solve pairs of simultaneous equations by substitution or elimination, see what the answer means on a graph, and spot when there is no solution.
- [How to Simplify Square Roots (Surds) Step by Step](https://mathsolving.space/blog/how-to-simplify-square-roots/): Simplify square roots like √50 and √72 by taking out square factors, combine surds, and rationalise denominators, with clear worked examples.
- [How to Simplify Fractions Step by Step](https://mathsolving.space/blog/how-to-simplify-fractions/): Simplify any fraction to its lowest terms using the greatest common factor, with worked examples, prime-factor checks and common mistakes.
- [How to Find the LCM and GCF (Three Methods)](https://mathsolving.space/blog/how-to-find-lcm-and-gcf/): Find the least common multiple and greatest common factor by listing, prime factorisation and Euclid's algorithm, and learn when to use each.
- [How to Check Your Own Maths Working (and Find the First Mistake)](https://mathsolving.space/blog/how-to-check-your-own-maths-working/): Simple habits to check your maths answers and spot exactly where a solution went wrong, with a worked example of finding the first wrong line.
- [How to Calculate Mean, Median and Mode](https://mathsolving.space/blog/how-to-calculate-mean-median-and-mode/): Find the mean, median, mode and range of a data set step by step, learn which average to use, and see how outliers affect each one.
- [How to Calculate a Percentage: 3 Easy Methods](https://mathsolving.space/blog/how-to-calculate-a-percentage/): Learn three simple ways to calculate a percentage of a number, find what percent one number is of another, and work out percentage change.
- [Derivatives for Beginners: Power, Product and Chain Rules](https://mathsolving.space/blog/derivatives-for-beginners/): Learn what a derivative is and how to differentiate with the power rule, product rule, quotient rule and chain rule, with worked examples.
- [Degrees and Radians: How to Convert and Why It Matters](https://mathsolving.space/blog/degrees-and-radians/): What a radian is, how to convert between degrees and radians in exact π form, a table of common angles, and why calculus uses radians.
- [Arithmetic vs Geometric Sequences: Formulas and Examples](https://mathsolving.space/blog/arithmetic-vs-geometric-sequences/): Tell arithmetic and geometric sequences apart, find the nth term and the sum of each, and see when a geometric series has a sum to infinity.
- [How to Find the Area of Common Shapes](https://mathsolving.space/blog/area-and-perimeter-of-common-shapes/): Area formulas for rectangles, triangles, circles, trapezoids and parallelograms, with worked examples, units explained and common mistakes.

## Pages

- [Blog](https://mathsolving.space/blog/): Step-by-step guides, worked examples and study tips.
- [Home](https://mathsolving.space/): Enter the numbers or expression, use the on-screen maths keypad, or tap an example.
- [Terms of Use](https://mathsolving.space/terms/): The terms for using this free maths website.
- [Privacy Policy](https://mathsolving.space/privacy/): What data this site collects (almost none) and how it is used.
- [Disclaimer](https://mathsolving.space/disclaimer/): How to use the results from this site, and their limits.
- [Cookie Policy](https://mathsolving.space/cookie-policy/): This site sets no cookies. Here is exactly what is stored on your device and why.
- [Contact](https://mathsolving.space/contact/): How to get in touch, report a problem or suggest a topic.
- [Accessibility](https://mathsolving.space/accessibility/): Our accessibility goals, what we have done, known limitations and how to report a barrier.
- [About](https://mathsolving.space/about/): Who this site is for, what it offers, and how it works.


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