All proofs

Every result proved on this site, in curriculum order. Each row links to the proof itself, not just the lesson holding it.

323 proofs · 179 lessons · 3 subjects

Pre-Algebra 80 proofs

Algebra I 106 proofs

1. The Language of Algebra

From Arithmetic to Algebra

Fractional Exponents and Radicals

Arithmetic with Expressions

Expanding and Factoring Expressions

Algebraic Fractions

2. Linear Equations

Solving Linear Equations

Linear Equations in Disguise

Word Problems with Linear Equations

Why consecutive integers are $n$ , $n + 1$ , $n + 2$

Solving for a Variable

3. Systems of Linear Equations

Equations with Two Variables

Why $ax + by = c$ has infinitely many solutions

Solving Systems by Substitution

Why replacing a variable preserves the solution set

Solving Systems by Elimination

Why combining the equations preserves the solution set

Word Problems with Systems

Why two independent conditions determine two unknowns

Systems with More Variables

Why eliminating a variable preserves the solution set and terminates

4. Ratios, Percents, and Proportion

Ratio Problems

Why one multiplier captures a whole ratio

Conversion Factors

Why multiplying by a conversion factor keeps the amount the same

Percent Problems

Why a percent change is one multiplication, and undoing it is one division

Direct and Inverse Proportion

One data pair fixes the whole relationship

Rate and Work Problems

5. Graphing Lines

Graphing Linear Equations

Why the solution points fall on a straight line

Slope

Why the slope does not depend on which two points you choose

Finding the Equation of a Line

Deriving point-slope form from the slope formula

Slope and Intercepts

Deriving slope-intercept form from point-slope form

Parallel and Perpendicular Lines

6. Inequalities

Inequality Basics

Why multiplying by a negative reverses the inequality

Solving Linear Inequalities

Why dividing by a negative reverses the symbol

Graphing Inequalities

Why one whole side satisfies the inequality

Introduction to Optimization

Why a linear objective is optimized at a corner

7. Quadratic Equations

Introduction to Quadratics

Why a product is zero only when a factor is zero

Factoring Quadratics

Why factoring $x^2 + bx + c$ means finding two numbers with sum $b$ and product $c$

Factoring Harder Quadratics

Why the factors of $ax^2 + bx + c$ satisfy $pq = a$ , $\ rs = c$ , and $\ ps + qr = b$

Sums and Products of Roots

8. Special Factorizations

Squares of Binomials

Difference of Squares

Why $a^2 - b^2 = (a + b)(a - b)$

Sum and Difference of Cubes

Why $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Rationalizing Denominators

Factoring by Grouping

9. Complex Numbers and the Quadratic Formula

Imaginary Numbers

Arithmetic with Complex Numbers

Completing the Square

Why adding $\left(\tfrac{b}{2}\right)^2$ turns $x^2 + bx$ into a perfect square

The Quadratic Formula

Deriving the quadratic formula from $ax^2 + bx + c = 0$

Applications of Quadratics

Why an area or product problem keeps exactly one root

10. Graphing Quadratics and Inequalities

Parabolas

The Equation of a Circle

Quadratic Inequalities

Why the sign is constant on each interval and flips at each root

Quadratic Optimization

Why the vertex holds the largest or smallest value

11. Functions

What Is a Function?

Why the vertical line test works

Combining and Composing Functions

Which inputs a composition accepts

Inverse Functions

Why an inverse needs a one-to-one function

Defining New Operations

The operation $a \oplus b = a + b + ab$ is commutative

12. Graphing Functions

Graphs of Functions

A point lies on the graph exactly when $y = f(x)$

Shifting Graphs

Stretching and Reflecting Graphs

Graphs of Inverse Functions

Swapping coordinates reflects a point across $y = x$

13. Polynomials, Exponentials, and Logarithms

Adding and Subtracting Polynomials

Why like terms combine and unlike terms do not

Multiplying Polynomials

Exponential Functions

Why $b^{x+y} = b^x \cdot b^y$

Compound Interest

Why annual compounding gives $A = P(1+r)^t$

Introduction to Logarithms

14. Special Functions

Radical Functions

Absolute Value Equations and Graphs

Floor and Ceiling

Why $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$

Rational Functions

Piecewise Functions

Why each input must belong to exactly one piece

15. Sequences and Series

Arithmetic Sequences

The $n$ th term is $a_n = a_1 + (n-1)d$

Arithmetic Series

Reverse and add: $2S_n = n(a_1 + a_n)$

Geometric Sequences

The $n$ th term is $a_n = a_1 r^{n-1}$

Geometric Series

The finite sum is $S_n = \frac{a_1(1 - r^{n})}{1 - r}$ for $r \ne 1$

Telescoping Sums

A telescoping sum collapses to $b_1 - b_{n+1}$

16. Special Manipulations

Raising Equations to Powers

Self-Similar Expressions

Why the value must satisfy $x = \sqrt{2 + x}$

Using Symmetry

Rewriting the standard symmetric expressions with $s$ and $p$ only

Algebra II 137 proofs

1. Equations and Inequalities

Solving Linear Equations

Literal Equations and Formulas

The three cases of $ax = b$ when $a$ and $b$ are parameters

Linear Inequalities

Absolute Value Equations and Inequalities

2. Functions and Their Graphs

Relations and Functions

The vertical line test decides whether $x$ is a legal input

Function Notation and Evaluation

Graphs of Functions

Transformations of Graphs

Composition of Functions

Inverse Functions

3. Systems of Equations and Inequalities

Systems in Two Variables

Systems in Three Variables

Why the solution set is a point, a line, a plane, or nothing

Systems of Inequalities

Matrices and Systems of Equations

Why the three row operations preserve the solution set

Determinants and Cramer's Rule

4. Quadratic Functions and Equations

Quadratic Functions and Parabolas

Solving Quadratics by Factoring

Completing the Square

The Quadratic Formula and the Discriminant

Quadratic Inequalities

Sum and Product of Roots

5. Complex Numbers

The Imaginary Unit and Complex Numbers

Arithmetic of Complex Numbers

The Complex Plane and Modulus

Complex Roots of Quadratics

6. Polynomial Division and Roots

Polynomials and Their Graphs

Polynomial Long Division

Synthetic Division and the Remainder Theorem

The Factor Theorem

The Rational Root Theorem

Proof of the Rational Root Theorem

7. Zeros of Polynomials

The Fundamental Theorem of Algebra

Complex and Irrational Roots

Vieta's Formulas

Vieta's formulas for a polynomial of degree $n$

Graphing Polynomial Functions

8. Rational Expressions and Functions

Simplifying Rational Expressions

Cancelling a common factor: $\dfrac{ac}{bc} = \dfrac{a}{b}$ whenever $b \neq 0$ and $c \neq 0$

Multiplying and Dividing Rational Expressions

Adding and Subtracting Rational Expressions

Solving Rational Equations

Clearing the denominators loses nothing and adds only excluded values

Graphs of Rational Functions

Why a factor that cancels leaves a hole, and one that survives forces a vertical asymptote

9. Radicals and Rational Exponents

Rational Exponents

Simplifying Radical Expressions

Operations with Radicals

$\sqrt a + \sqrt b = \sqrt{a+b}$ holds only when $a = 0$ or $b = 0$

Solving Radical Equations

Rationalizing and Radical Conjugates

10. Exponential and Logarithmic Functions

Exponential Functions and Graphs

Introduction to Logarithms

$b^x$ is one-to-one, so it has an inverse

Properties of Logarithms

Solving Exponential and Logarithmic Equations

Exponential Growth and Decay

After $t$ steps, $A(t) = P(1+r)^t$

The Natural Base e

Compounding ever more finely turns $P\left(1 + \frac{r}{n}\right)^{nt}$ into $Pe^{rt}$

11. Sequences and Series

Sequences and Notation

The explicit rule $a_n = n^2 + 1$ and the recursion $a_1 = 2$ , $a_n = a_{n-1} + 2n - 1$ define the same sequence

Arithmetic Sequences and Series

Geometric Sequences and Series

Infinite Geometric Series

If $|r| < 1$ , then $a + ar + ar^2 + \cdots = \dfrac{a}{1-r}$

The Binomial Theorem

12. Conic Sections

Parabolas

Circles

Ellipses

The distance condition is exactly $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Hyperbolas

The distance condition forces $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

13. Probability and Statistics

Counting Principles, Permutations, and Combinations

Introduction to Probability

Data Distributions and Statistics

The Normal Distribution

All proofs

Pre-Algebra 80 proofs

1. Foundations of Arithmetic

2. Integers and the Number Line

3. Factors and Multiples

4. Fractions

5. Decimals

6. Exponents and Roots

7. Ratios, Rates, and Percents

8. Introduction to Algebra

9. Geometry Basics

10. Data, Counting, and Probability

Algebra I 106 proofs

1. The Language of Algebra

2. Linear Equations

3. Systems of Linear Equations

4. Ratios, Percents, and Proportion

5. Graphing Lines

6. Inequalities

7. Quadratic Equations

8. Special Factorizations

9. Complex Numbers and the Quadratic Formula

10. Graphing Quadratics and Inequalities

11. Functions

12. Graphing Functions

13. Polynomials, Exponentials, and Logarithms

14. Special Functions

15. Sequences and Series

16. Special Manipulations

Algebra II 137 proofs

1. Equations and Inequalities

2. Functions and Their Graphs

3. Systems of Equations and Inequalities

4. Quadratic Functions and Equations

5. Complex Numbers

6. Polynomial Division and Roots

7. Zeros of Polynomials

8. Rational Expressions and Functions

9. Radicals and Rational Exponents

10. Exponential and Logarithmic Functions

11. Sequences and Series

12. Conic Sections

13. Probability and Statistics

Geometry coming soon

Precalculus coming soon

Calculus AB coming soon

Calculus BC coming soon

Statistics and Probability coming soon