Algebra guides
Solve equations, work with expressions, and master algebraic concepts with detailed explanations.
19 guides · each with a free step-by-step calculator
Algebra Solver: Master Equations & Expressions with Step-by-Step Solutions
Think of an equation as a balance scale: whatever you do to one side, you must do to the other. That one idea solves almost every algebra problem you'll meet in school. In this guide you'll learn how to solve linear equations, pairs of equations and quadratics, step by step.
Quadratic Formula Explained: Solve Any Quadratic Equation Step by Step
You throw a ball in the air. When will it be exactly 15 metres high? The answer is hidden in a quadratic equation, and one formula can crack every one of them. In this guide you'll learn the quadratic formula, what the discriminant tells you, and how to use it without the usual sign mistakes.
How to Find the Determinant of a 2×2 Matrix
Can one number tell you whether a whole matrix can be undone? Yes, and for a 2×2 matrix it takes just two multiplications and a subtraction. In this guide you'll learn the ad − bc rule, what the answer means as an area, and why 0 is a warning sign.
How to Find Eigenvalues of a 2×2 Matrix
When a matrix stretches and turns space, are there any directions it leaves pointing the same way? Usually yes, and eigenvalues tell you how much those directions stretch. This guide shows you how to find both eigenvalues of a 2×2 matrix using just the trace and the determinant.
How to Multiply 2×2 Matrices (Step-by-Step)
Here's a surprise: with matrices, A × B is usually not the same as B × A. Matrix multiplication follows its own rule, row times column. This guide shows you that rule for 2×2 matrices, with four short calculations and a quick way to check your work.
How to Find the Inverse of a 2×2 Matrix
Dividing by 5 undoes multiplying by 5. But you can't divide by a matrix, so how do you undo one? You use its inverse. For a 2×2 matrix there is a quick recipe: swap, change signs, divide. This guide walks you through it and shows how to check your answer.
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How to Find the Determinant of a 3×3 Matrix
Nine numbers, one answer. The determinant of a 3×3 matrix tells you whether three equations have exactly one solution, and how much a 3D shape grows or shrinks. This guide shows you the cover-up method that turns it into three easy 2×2 determinants.
Dot Product of Two Vectors Explained Simply
Every time you add up a shopping bill, quantity times price for each item, you are doing a dot product. The same simple sum tells you whether two arrows point the same way or meet at a right angle. This guide shows you how to calculate it and what the answer means.
Cross Product of Vectors and Its Magnitude
How do you find an arrow that points straight out of a tilted surface, or the area of a slanted triangle in 3D? The cross product does both. This guide shows you how to calculate it, find its length, and use the right-hand rule to get its direction.
How to Find the Magnitude of a Vector
A drone flies 2 m east, 3 m north and 6 m up. How far is it from where it started? Not 11 m. The straight-line answer is the vector's magnitude, and it comes from Pythagoras. This guide shows you how to find the length of any 2D or 3D vector.
How to Find the Slope of a Line (Step-by-Step)
Why do some wheelchair ramps feel gentle while others feel like a wall? It's all down to one number: slope. Give this guide two points and you'll be able to find the slope of any line in under a minute.
What Is a Logarithm? Formula and Examples
How can an earthquake that's "only" one point stronger on a scale actually be ten times more powerful? The answer is logarithms. This guide shows you exactly what log₁₀(x) means and how to find it.
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How to Calculate Exponential Growth (Formula)
How does a single bacteria colony turn into billions in under a day? It multiplies instead of adds. This guide breaks down the exponential growth formula so you can predict that kind of runaway growth yourself.
How to Solve a Linear Equation for x
Your taxi fare comes to $20, and you know the flat fee and the rate per kilometer — but how far did you actually go? A linear equation is hiding that answer. Unlock it in four simple steps.
How to Find the nth Term of an Arithmetic Sequence
Need to know how many seats are in row 12 of a theatre without counting rows 1 through 11 first? An arithmetic sequence lets you jump straight to any term with one short formula.
How to Find the Sum of an Arithmetic Sequence
Adding up dozens of seat counts row by row is slow and easy to get wrong. If every row grows by the same fixed amount, there's a shortcut formula that totals them all in one calculation.
How to Find the nth Term of a Geometric Sequence
How many new people see a video by round 6 if the number of viewers doubles every round? A geometric sequence multiplies instead of adds, and this formula finds any term instantly without multiplying step by step.
How to Find the Sum of a Geometric Series
A video that doubles its new viewers every round doesn't just have big single rounds — the running total across all rounds explodes even faster. This formula adds every term of a geometric series in one calculation.
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How to Solve Simultaneous Equations (2 Variables)
Two apples and a banana cost $11 total, and an apple costs a dollar more than a banana — so what does each fruit actually cost? Simultaneous equations pin down two unknowns at once, and Cramer's rule gets you there directly.
