Algebra Cheat Sheet
Algebra is the backbone of modern mathematics and a crucial tool for problem-solving in many fields. Our Algebra Cheat Sheet is designed to simplify complex topics, from the distributive property and quadratic formula to factoring and exponent rules. Whether you’re grappling with linear equations, exploring binomial expansions, or mastering radical simplification, this guide provides clear, concise explanations and practical examples that help demystify each concept. Perfect for students, educators, and anyone needing a quick refresher, this cheat sheet turns challenging algebra topics into manageable steps.
In addition, the cheat sheet serves as an invaluable resource for both classroom learning and independent study. With detailed explanations, step-by-step examples, and a neatly organized layout, you can quickly find the formulas and methods you need to solve problems confidently. Use it for homework support, test preparation, or as a handy reference during lectures. Download, print, and keep this essential guide by your side as you unlock the world of algebra and build a strong foundation in mathematics.
| Concept | Formula/Example | Notes |
|---|---|---|
| Distributive Property | a(b + c) = ab + ac | Expands expressions |
| Quadratic Formula | x = (-b ± √(b² - 4ac))/(2a) | Solves quadratic equations |
| Slope-Intercept Form | y = mx + b | Linear equations |
| Point-Slope Form | y - y₁ = m(x - x₁) | Line given a point and slope |
| Factoring (Difference of Squares) | x² - 9 = (x - 3)(x + 3) | Basic factorization |
| Exponent Rule | a^m · a^n = a^(m+n) | Add exponents with same base |
| Radical Simplification | √(ab) = √a · √b | Simplifies square roots |
| Logarithm Product Rule | log(ab) = log a + log b | Splits logarithms |
| Absolute Value | |x| | Distance from zero |
| Binomial Theorem (Square) | (a + b)² = a² + 2ab + b² | Expands squared binomials |
| Power of a Power | (a^m)^n = a^(m·n) | Multiply exponents |
| Difference of Cubes | a³ - b³ = (a - b)(a² + ab + b²) | Factorization pattern |
| Factoring Trinomials | x² + 5x + 6 = (x + 2)(x + 3) | For factorable quadratics |
| Completing the Square | x² + 6x + 9 = (x + 3)² | Rewrite quadratics as perfect squares |
| Simplifying Rational Expressions | (x² - 9)/(x - 3) = x + 3 | Cancel common factors |
| System of Equations |
2x + 3y = 6 x - y = 2 |
Solve using substitution/elimination |
| Properties of Equality | If a = b, then a + c = b + c | Operations on both sides maintain equality |
| Exponent Quotient Rule | a^m / a^n = a^(m - n) | Subtract exponents with same base |
| Term | Explanation |
|---|---|
| Distributive Property | The distributive property allows you to multiply a single term by each term inside a parenthesis. For example, in a(b + c) = ab + ac, the term a multiplies both b and c. This is essential for expanding expressions and simplifying algebraic equations. |
| Quadratic Formula | The quadratic formula, x = (-b ± √(b² - 4ac))/(2a), solves any quadratic equation of the form ax² + bx + c = 0. By substituting values for a, b, and c, you can find the two solutions for x. It is a powerful tool for finding the roots of quadratic functions. |
| Slope-Intercept Form | In the equation y = mx + b, m is the slope and b is the y-intercept. This form of a linear equation makes it simple to graph a line by identifying its steepness (m) and where it crosses the y-axis (b). It is one of the most common forms used in coordinate geometry. |
| Point-Slope Form | The point-slope form, y - y₁ = m(x - x₁), is useful when you know a point (x₁, y₁) on the line and its slope m. It provides a straightforward method to write the equation of a line and is often used in problems that involve determining a line’s equation from limited information. |
| Factoring (Difference of Squares) | The difference of squares is a special factoring pattern. For any two numbers a and b, a² - b² can be factored into (a - b)(a + b). For example, x² - 9 factors into (x - 3)(x + 3). This technique is frequently used to simplify algebraic expressions. |
| Exponent Rule | When multiplying terms with the same base, you add their exponents. For example, a^m · a^n equals a^(m+n). This rule simplifies expressions involving powers and is one of the fundamental properties of exponents. |
| Radical Simplification | Radicals such as square roots can often be broken down by expressing the number under the radical as a product. For instance, √(ab) = √a · √b. This helps in simplifying complex radical expressions by reducing them to simpler factors. |
| Logarithm Rules | Logarithms transform multiplication into addition. The product rule, log(ab) = log a + log b, is particularly useful for simplifying logarithmic expressions. Other rules include the quotient and power rules, which together make logarithms a powerful tool in algebra. |
| Absolute Value | The absolute value of a number, denoted by |x|, is its distance from zero on the number line, regardless of direction. For example, |−5| = 5. This concept is important for understanding distance and for solving equations and inequalities that involve absolute values. |
| Binomial Theorem | The binomial theorem provides a formula for expanding expressions raised to a power. For example, (a + b)² expands to a² + 2ab + b². More generally, it gives the coefficients for each term in the expansion of (a + b)^n, which is very useful in both algebra and probability. |
| Power of a Power Rule | This rule states that when you raise an exponent to another exponent, you multiply the exponents: (a^m)^n = a^(m·n). This rule is essential for simplifying expressions that involve multiple layers of exponents. |
| Difference of Cubes | The difference of cubes formula is a³ - b³ = (a - b)(a² + ab + b²). This factorization technique is used to simplify cubic expressions and solve equations involving cubic terms. |
| Factoring Trinomials | Many quadratic expressions in the form x² + bx + c can be factored into two binomials if there exist two numbers that add to b and multiply to c. For example, x² + 5x + 6 factors into (x + 2)(x + 3). This is a key method for solving quadratic equations. |
| Completing the Square | Completing the square is a technique used to rewrite a quadratic equation in the form (x + d)² = e. For example, x² + 6x + 9 can be written as (x + 3)². This method is particularly useful for solving quadratic equations and analyzing parabolic graphs. |
| Simplifying Rational Expressions | Rational expressions are fractions where the numerator and/or denominator are polynomials. They can often be simplified by factoring and canceling common terms. For example, (x² - 9)/(x - 3) simplifies to x + 3, provided x ≠ 3. |
| Systems of Equations | A system of linear equations consists of two or more equations with the same variables. Techniques such as substitution and elimination are used to find the values of the variables that satisfy all equations simultaneously. For example, solving 2x + 3y = 6 and x - y = 2 together yields a unique solution for x and y. |
| Properties of Equality | The properties of equality state that performing the same operation on both sides of an equation maintains the equality. For instance, if a = b, then a + c = b + c and a - c = b - c. These properties are foundational in solving algebraic equations. |
| Exponent Quotient Rule | When dividing like bases, subtract the exponent of the denominator from that of the numerator: a^m / a^n = a^(m - n). This rule simplifies expressions where division of powers occurs. |