📐 Complete Algebra Cheat Sheet
Table of Contents & Formula Reference Guide
For IB, AP, GCSE, IGCSE & More
📚 Table of Contents
1️⃣ Sequences (Arithmetic & Geometric)
2️⃣ Exponents & Powers
3️⃣ Logarithms & Laws
4️⃣ Binomial Expansion
5️⃣ Quadratic Formulas
6️⃣ Polynomial Identities
7️⃣ Linear Equations & Functions
8️⃣ Inequalities
🎯 Why This Algebra Cheat Sheet Matters
Algebra forms the foundation of advanced mathematics across all major curricula including IB (International Baccalaureate), AP (Advanced Placement), GCSE, IGCSE, A-Levels, and SAT preparation. This comprehensive formula sheet consolidates essential algebraic concepts, providing students with quick access to critical formulas during revision and problem-solving sessions.
- Universal Application: These formulas apply across multiple examination boards and curricula
- Exam Essential: Core formulas frequently tested in standardized assessments
- Time-Saving: Quick reference guide reduces formula lookup time during practice
- Comprehensive Coverage: From basic sequences to advanced binomial expansion
1️⃣ Sequences
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms.
nth Term: un = u1 + (n - 1)d
Sum of n Terms: Sn = (n/2)(2u1 + (n - 1)d)
Alternative Sum Formula: Sn = (n/2)(u1 + un)
Where: u1 = first term, d = common difference, n = number of terms
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms.
nth Term: un = u1 · rn-1
Sum of n Terms: Sn = u1(1 - rn)/(1 - r), when r ≠ 1
Sum to Infinity: S∞ = u1/(1 - r), when |r| < 1
Where: u1 = first term, r = common ratio, n = number of terms
Sigma Notation
A shorthand notation to show the sum of a number of terms in a sequence.
Σ notation represents summation: Σ (from n=1 to k) of an
Example: Σ (from n=1 to 10) of 3n = 3(1) + 3(2) + 3(3) + ... + 3(10) = 165
💡 Key Insight: Arithmetic sequences have constant differences (linear growth), while geometric sequences have constant ratios (exponential growth). Recognizing sequence type is crucial for applying the correct formula.
2️⃣ Exponents & Powers
Basic Exponent Laws
x1 = x
x0 = 1
xm · xn = xm+n
xm / xn = xm-n
(xm)n = xm·n
(x · y)n = xn · yn
Negative & Fractional Exponents
x-1 = 1/x
x-n = 1/xn
x1/2 = √x
x1/n = n√x
Advanced Exponent Rules
xm/n = n√(xm)
x-m/n = 1/xm/n
√x · √y = √(xy)
n√xm = xm/n
⚠️ Common Mistake: Remember that xm + xn ≠ xm+n. The exponent laws only apply to multiplication and division, not addition or subtraction.
3️⃣ Logarithms & Laws
Basic Logarithm Definitions
Fundamental Definition: loga(x) = y ⟺ ay = x
Identity: loga(ax) = x
Inverse: aloga(x) = x
Let ax = b, isolate x from the exponent: x = loga(b)
Let loga(x) = b, isolate x from the logarithm: x = ab
Laws of Logarithms
1. Product Rule: logc(a) + logc(b) = logc(a · b)
2. Quotient Rule: logc(a) - logc(b) = logc(a/b)
3. Power Rule: n·logc(a) = logc(an)
4. Change of Base: logb(a) = logc(a) / logc(b)
Special Logarithms
Common Log: log(x) = log10(x)
Base 10 logarithm (calculator log button)
Natural Log: ln(x) = loge(x)
Base e logarithm (e ≈ 2.71828...)
🎓 Pro Tip: When solving logarithmic equations, always check your solutions. Logarithms are only defined for positive arguments, so solutions that make any logarithm argument negative or zero must be rejected.
4️⃣ Binomial Expansion
Binomial Theorem
In an expansion of a binomial in the form (a + b)n, each term can be described as nCr·an-r·br, where nCr is the coefficient.
(a + b)n = nC0·an·b0 + nC1·an-1·b1 + nC2·an-2·b2 + ... + nCn·a0·bn
Binomial Coefficient
nCr = n! / (r!(n-r)!)
Also written as ⁿCᵣ or (n choose r) or C(n,r)
Factorial: n! = n × (n-1) × (n-2) × ... × 2 × 1, and 0! = 1
Pascal's Triangle Method
Find the coefficient using Pascal's Triangle:
n = 0: 1
n = 1: 1 1
n = 2: 1 2 1
n = 3: 1 3 3 1
n = 4: 1 4 6 4 1
n = 5: 1 5 10 10 5 1
Each number is the sum of the two numbers directly above it
Common Binomial Expansions
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
📝 Calculator Tip: Most scientific calculators have an nCr function (combination function) to quickly calculate binomial coefficients. This is especially useful for larger values of n.
5️⃣ Quadratic Formulas
Standard Form & Quadratic Formula
For a quadratic equation in standard form:
Standard Form: ax2 + bx + c = 0
Quadratic Formula: x = (-b ± √(b2 - 4ac)) / (2a)
Discriminant
Discriminant: Δ = b2 - 4ac
If Δ > 0: Two distinct real roots
If Δ = 0: One repeated real root (perfect square)
If Δ < 0: No real roots (two complex roots)
Vertex Form
y = a(x - h)2 + k
Vertex is at point (h, k)
Opens upward if a > 0, downward if a < 0
Completing the Square
To convert from standard form to vertex form:
ax2 + bx + c = a(x + b/(2a))2 + (c - b2/(4a))
Vertex: h = -b/(2a), k = c - b2/(4a)
Sum and Product of Roots
Sum of Roots: α + β = -b/a
Product of Roots: α × β = c/a
🎯 Strategy: Use factoring when possible for simple quadratics. Use the quadratic formula for complex coefficients or when factoring is difficult. Completing the square is useful when finding the vertex or for deriving the quadratic formula.
6️⃣ Polynomial Identities
Difference & Sum of Squares
a2 - b2 = (a - b)(a + b)
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
a2 + b2 = (a + b)2 - 2ab
Cubic Identities
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
Three-Term Expansions
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca
Fourth Power Identities
a4 - b4 = (a - b)(a + b)(a2 + b2)
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a - b)4 = a4 - 4a3b + 6a2b2 - 4ab3 + b4
💡 Recognition Tip: These identities are crucial for factoring complex expressions. Look for patterns like difference of squares, perfect square trinomials, and sum/difference of cubes to simplify algebraic expressions efficiently.
7️⃣ Linear Equations & Functions
Slope (Gradient)
Slope Formula: m = (y2 - y1) / (x2 - x1) = rise/run
For a line passing through points (x1, y1) and (x2, y2)
Equation Forms
Slope-Intercept Form: y = mx + b
m = slope, b = y-intercept
Point-Slope Form: y - y1 = m(x - x1)
Passes through point (x1, y1) with slope m
Standard Form: Ax + By = C
A, B, and C are integers
Parallel & Perpendicular Lines
Parallel Lines: m1 = m2
Same slope
Perpendicular Lines: m1 × m2 = -1
Negative reciprocal slopes
Distance & Midpoint
Distance Formula: d = √((x2 - x1)2 + (y2 - y1)2)
Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
8️⃣ Inequalities
Inequality Rules
Basic properties of inequalities:
If a < b, then a + c < b + c (Addition Property)
If a < b, then a - c < b - c (Subtraction Property)
If a < b and c > 0, then ac < bc (Multiplication by Positive)
If a < b and c < 0, then ac > bc (Multiplication by Negative - REVERSES)
Absolute Value Inequalities
|x| < a means -a < x < a
|x| > a means x < -a or x > a
Where a > 0
⚠️ Critical Rule: When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed. This is one of the most common errors in algebra.
📖 Study Tips for Mastering Algebra
- Practice Regularly: Algebra skills improve with consistent practice. Work through varied problems daily rather than cramming.
- Understand, Don't Memorize: Focus on understanding why formulas work rather than just memorizing them.
- Check Your Work: Always substitute solutions back into original equations to verify correctness.
- Work Backwards: When stuck, try working from the answer choices backward (especially useful in multiple-choice exams).
- Identify Patterns: Recognize common algebraic patterns like difference of squares, perfect square trinomials, and factoring techniques.
- Use Multiple Methods: Learn different approaches to solving problems (graphing, algebraic, numeric) to deepen understanding.
- Master the Basics First: Ensure strong fundamentals in order of operations, fraction operations, and basic equation solving before advancing.
🎓 Common Algebra Applications Across Curricula
- IB Mathematics: Algebra forms the foundation for Analysis & Approaches (AA) and Applications & Interpretation (AI) courses, appearing in Paper 1 and Paper 2 assessments.
- AP Calculus: Strong algebraic manipulation skills are essential for differentiation, integration, and solving calculus problems efficiently.
- GCSE/IGCSE: Algebra comprises approximately 30% of exam content, including solving equations, rearranging formulas, and graph work.
- A-Level Mathematics: Advanced algebra including partial fractions, mathematical induction, and complex algebraic manipulation.
- SAT/ACT: Approximately 35-40% of math questions involve algebraic reasoning and equation-solving.
💪 Master these formulas through practice and understanding!
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1.1 Sequences
Arithmetic: +/− common difference
with u1 = a = 1st term, d = common difference.
Geometric: ×/÷ common ratio
with u1 = a = 1st term, r = common ratio.
Sigma notation
A shorthand to show the sum of a number of terms in a sequence.
1.2 Exponents and logarithms
Exponents
Logarithms
loga ax = x
aloga b = b
Let ax = b, isolate x from the exponent: loga ax = x = loga b
Let loga x = b, isolate x from the logarithm: aloga x = x = ab
Laws of logarithms
1.3. Binomial Expansion
In an expansion of a binomial in the form (a + b)n . Each term can be described as nCran−rbr, where nCr is the coefficient.
The full expansion can be written thus
Find the coefficient using either pascals triangle
Or the nCr function on your calculator