Complete formula reference with micro-examples for quick contest lookup. Use this as your go-to resource during practice and contests.
🗂️ Table of Contents#
🔢 Basic Algebra#
Master these fundamental patterns first—they form the foundation for all advanced algebra!
Factoring Patterns#
These factoring patterns appear in 80% of AMC algebra problems!
Order of Operations#
Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
Example Walkthrough:
$$2 + 3 \times 4^2 = 2 + 3 \times 16 = 2 + 48 = 50$$
Step-by-step:
- Exponents first: $4^2 = 16$
- Multiplication: $3 \times 16 = 48$
- Addition: $2 + 48 = 50$
🎯 Quadratics#
Quadratic problems are high-frequency in AMC contests. Master both the formula and vertex form!
| Concept | Formula | Micro-Example | Usage |
|---|
| Quadratic formula | $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ | $x^2 - 5x + 6 = 0$: $x = \frac{5 \pm \sqrt{25-24}}{2} = 2, 3$ | When factoring fails |
| Discriminant | $\Delta = b^2 - 4ac$ | $x^2 - 4x + 4 = 0$: $\Delta = 16 - 16 = 0$ (one root) | Quick root analysis |
| Nature of roots | $\Delta > 0$: 2 real, $\Delta = 0$: 1 real, $\Delta < 0$: 2 complex | $x^2 + 1 = 0$: $\Delta = -4 < 0$ (no real roots) | Problem classification |
Vertex form makes it easy to find the vertex, axis of symmetry, and transformations!
| Concept | Formula | Micro-Example | Key Insight |
|---|
| Vertex form | $y = a(x-h)^2 + k$ | $y = 2(x-3)^2 + 1$ has vertex $(3,1)$ | Vertex at $(h,k)$ |
| Vertex coordinates | $h = -\frac{b}{2a}$, $k = \frac{4ac-b^2}{4a}$ | $y = x^2 - 4x + 3$: vertex $(2,-1)$ | Convert from standard form |
| Axis of symmetry | $x = -\frac{b}{2a}$ | $y = x^2 - 4x + 3$: axis $x = 2$ | Always passes through vertex |
🧮 Polynomials#
Vieta’s formulas let you find relationships between roots without solving the equation!
| Degree | Sum of Roots | Product of Roots | Micro-Example | Contest Usage |
|---|
| Quadratic $ax^2 + bx + c$ | $r_1 + r_2 = -\frac{b}{a}$ | $r_1 \cdot r_2 = \frac{c}{a}$ | $x^2 - 5x + 6 = 0$: sum = 5, product = 6 | Very common |
| Cubic $ax^3 + bx^2 + cx + d$ | $r_1 + r_2 + r_3 = -\frac{b}{a}$ | $r_1 \cdot r_2 \cdot r_3 = -\frac{d}{a}$ | $x^3 - 6x^2 + 11x - 6 = 0$: sum = 6, product = 6 | AMC 12 |
Remainder & Factor Theorems#
Use these theorems to quickly test if a linear factor divides a polynomial!
| Theorem | Statement | Micro-Example | When to Use |
|---|
| Remainder | Remainder when $f(x) \div (x-a)$ is $f(a)$ | $f(x) = x^3 + 2x^2 - 5x + 1$: remainder when divided by $(x-2)$ is $f(2) = 7$ | Finding remainders |
| Factor | $(x-a)$ is a factor of $f(x)$ if and only if $f(a) = 0$ | If $f(3) = 0$, then $(x-3)$ divides $f(x)$ | Factoring polynomials |
📊 Series & Sequences#
Sequences and series are high-frequency topics. Master both arithmetic and geometric patterns!
Arithmetic Sequences#
| Concept | Formula | Micro-Example | When to Use |
|---|
| $n$th term | $a_n = a_1 + (n-1)d$ | $a_5 = 3 + 4 \cdot 2 = 11$ | Finding specific terms |
| Sum (form 1) | $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ | $S_{10} = 5(6 + 9 \cdot 2) = 120$ | When you know $a_1$ and $d$ |
| Sum (form 2) | $S_n = \frac{n}{2}(a_1 + a_n)$ | $S_{10} = 5(3 + 21) = 120$ | When you know first and last terms |
Geometric Sequences#
| Concept | Formula | When to Use |
|---|
| $n$th term | $a_n = a_1 \cdot r^{n-1}$ | Finding specific terms |
| Sum of first $n$ terms | $S_n = a_1 \frac{1-r^n}{1-r}$ | Finite geometric series |
| Infinite sum (if $|r| < 1$) | $S_\infty = \frac{a_1}{1-r}$ | Convergent series |
🔢 Exponents & Logarithms#
These rules are essential for simplifying complex expressions and solving exponential equations!
Exponent Rules#
| Rule | Formula | Micro-Example | Common Mistake |
|---|
| Product | $a^x \cdot a^y = a^{x+y}$ | $2^3 \cdot 2^4 = 2^7 = 128$ | Don’t add bases! |
| Power | $(a^x)^y = a^{xy}$ | $(3^2)^3 = 3^6 = 729$ | Don’t multiply exponents! |
| Quotient | $\frac{a^x}{a^y} = a^{x-y}$ | $\frac{5^7}{5^3} = 5^4 = 625$ | Subtract exponents! |
| Zero exponent | $a^0 = 1$ (for $a \neq 0$) | $7^0 = 1$ | Any non-zero to power 0 = 1 |
| Negative exponent | $a^{-x} = \frac{1}{a^x}$ | $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ | Move to denominator! |
Logarithm Rules#
Logarithms are the inverse of exponentials. Use these rules to simplify log expressions!
| Rule | Formula | Micro-Example | Key Insight |
|---|
| Product | $\log_a(xy) = \log_a x + \log_a y$ | $\log_2(8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5$ | Log of product = sum of logs |
| Quotient | $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$ | $\log_3\left(\frac{27}{9}\right) = \log_3 27 - \log_3 9 = 3 - 2 = 1$ | Log of quotient = difference of logs |
| Power | $\log_a(x^y) = y \log_a x$ | $\log_3(9^2) = 2 \log_3 9 = 2 \cdot 2 = 4$ | Exponent becomes coefficient |
| Change of base | $\log_a x = \frac{\log_b x}{\log_b a}$ | $\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} = \frac{0.903}{0.301} = 3$ | Convert to any base |
⚖️ Inequalities#
These inequalities are crucial for AMC 12 problems and optimization questions!
Basic Inequalities#
| Concept | Formula | Micro-Example | When to Use |
|---|
| AM-GM (2 variables) | $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b > 0$ | $\frac{4+9}{2} = 6.5 \geq \sqrt{36} = 6$ | Optimization problems |
| AM-GM (3 variables) | $\frac{a+b+c}{3} \geq \sqrt[3]{abc}$ for $a,b,c > 0$ | $\frac{2+3+6}{3} = \frac{11}{3} \geq \sqrt[3]{36} \approx 3.3$ | 3-variable optimization |
| Cauchy-Schwarz | $(a^2+b^2)(c^2+d^2) \geq (ac+bd)^2$ | $(1^2+2^2)(3^2+4^2) = 5 \cdot 25 = 125 \geq (3+8)^2 = 121$ | Vector inequalities |
🧮 Complex Numbers#
Complex numbers extend real numbers and are essential for advanced algebra!
Basic Operations#
| Concept | Formula | Micro-Example | Key Insight |
|---|
| Imaginary unit | $i^2 = -1$ | $i^3 = i^2 \cdot i = -i$ | Powers cycle: $i, -1, -i, 1$ |
| Complex conjugate | $\overline{a+bi} = a-bi$ | $\overline{3+4i} = 3-4i$ | Conjugate flips sign of imaginary part |
| Modulus | $|a+bi| = \sqrt{a^2+b^2}$ | $|3+4i| = \sqrt{9+16} = 5$ | Distance from origin |
| Product with conjugate | $(a+bi)(a-bi) = a^2+b^2$ | $(3+4i)(3-4i) = 9+16 = 25$ | Always gives real number |
You now have a comprehensive formula reference! Practice regularly and use this as your go-to resource during contests.
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