Core Concepts & Key Formulas
Algebra — Equations & Expressions
TOPIC 01 · ALGEBRA
Quadratic Formula: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Vieta's Formulas: \(r_1+r_2 = -\dfrac{b}{a},\quad r_1 r_2 = \dfrac{c}{a}\)
Vieta's Formulas: \(r_1+r_2 = -\dfrac{b}{a},\quad r_1 r_2 = \dfrac{c}{a}\)
📌 Must Memorize
- Discriminant \(\Delta = b^2-4ac\): \(>0\) two real roots, \(=0\) double root, \(<0\) complex
- Sum of roots \(= -b/a\), Product of roots \(= c/a\)
- \((a+b)^2 = a^2+2ab+b^2\), \((a-b)^2 = a^2-2ab+b^2\)
- \(a^2-b^2 = (a+b)(a-b)\)
🔑 Quick Example
If \(x^2 - 5x + 6 = 0\), find the sum of roots.
By Vieta: sum \(= 5\), product \(= 6\). Roots are \(2\) and \(3\). ✓
Answer: 5
Number Theory — Divisibility & Primes
TOPIC 02 · NUMBER THEORY
\(\gcd(a,b) \times \text{lcm}(a,b) = a \times b\)
Number of divisors of \(n = p_1^{a_1} p_2^{a_2}\cdots\): \(\tau(n)=(a_1+1)(a_2+1)\cdots\)
Number of divisors of \(n = p_1^{a_1} p_2^{a_2}\cdots\): \(\tau(n)=(a_1+1)(a_2+1)\cdots\)
📌 Must Memorize
- Divisibility rules: 2 (even), 3 (digit sum ÷3), 9 (digit sum ÷9), 11 (alternating sum)
- Primes to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
- Every integer \(n>1\) has a unique prime factorization
- Remainder theorem: \(a \equiv r \pmod{m}\)
🔑 Quick Example
How many positive divisors does \(360\) have?
\(360 = 2^3 \cdot 3^2 \cdot 5\), so \(\tau(360) = (3+1)(2+1)(1+1) = 24\)
Answer: 24
Geometry — Triangles, Circles & Area
TOPIC 03 · GEOMETRY
Heron's: \(A = \sqrt{s(s-a)(s-b)(s-c)}\), \(s=\tfrac{a+b+c}{2}\)
Circle: Area \(= \pi r^2\), Circumference \(= 2\pi r\)
Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Circle: Area \(= \pi r^2\), Circumference \(= 2\pi r\)
Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
📌 Must Memorize
- Area of triangle: \(\frac{1}{2}bh\), also \(\frac{1}{2}ab\sin C\)
- Similar triangles: ratios of sides are equal, areas scale as ratio squared
- Inscribed angle = half the central angle subtending the same arc
- 30-60-90 sides: \(1 : \sqrt{3} : 2\); 45-45-90: \(1:1:\sqrt{2}\)
🔑 Quick Example
A right triangle has legs 6 and 8. Find the hypotenuse.
\(\sqrt{6^2+8^2} = \sqrt{36+64} = \sqrt{100} = 10\)
Answer: 10
Combinatorics — Counting & Probability
TOPIC 04 · COMBINATORICS
\(\displaystyle\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
Permutations: \(P(n,k) = \dfrac{n!}{(n-k)!}\)
Probability: \(P(A) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}\)
Permutations: \(P(n,k) = \dfrac{n!}{(n-k)!}\)
Probability: \(P(A) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}\)
📌 Must Memorize
- Complement: \(P(A') = 1 - P(A)\)
- Stars and bars: distributing \(n\) into \(k\) bins: \(\binom{n+k-1}{k-1}\)
- Pascal's identity: \(\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\)
- Inclusion-Exclusion: \(|A\cup B|=|A|+|B|-|A\cap B|\)
🔑 Quick Example
How many ways to choose 2 students from 5?
\(\binom{5}{2} = \frac{5!}{2!\cdot3!} = 10\)
Answer: 10
Functions, Sequences & Series
TOPIC 05 · FUNCTIONS
Arithmetic sequence: \(a_n = a_1 + (n-1)d\), \(S_n = \dfrac{n(a_1+a_n)}{2}\)
Geometric sequence: \(a_n = a_1 r^{n-1}\), \(S_n = \dfrac{a_1(1-r^n)}{1-r}\)
Geometric sequence: \(a_n = a_1 r^{n-1}\), \(S_n = \dfrac{a_1(1-r^n)}{1-r}\)
📌 Must Memorize
- Floor \(\lfloor x \rfloor\): greatest integer \(\le x\)
- For arithmetic: common difference \(d\) is constant
- For geometric: common ratio \(r\) is constant
- Infinite geometric sum: \(S_\infty = \dfrac{a_1}{1-r}\) for \(|r|<1\)
🔑 Quick Example
Find the sum of the first 10 terms of \(1, 3, 5, 7, \ldots\)
\(S_{10} = \frac{10(1+19)}{2} = 100\)
Answer: 100
Rates, Ratios & Proportions
TOPIC 06 · RATES
Distance \(= \text{Rate} \times \text{Time}\)
Combined work: \(\dfrac{1}{T} = \dfrac{1}{t_1} + \dfrac{1}{t_2}\)
Combined work: \(\dfrac{1}{T} = \dfrac{1}{t_1} + \dfrac{1}{t_2}\)
📌 Must Memorize
- Mixture problems: weighted average formula
- Relative speed: same direction \(|v_1-v_2|\), opposite \(v_1+v_2\)
- Percentage change: \(\dfrac{\text{new}-\text{old}}{\text{old}} \times 100\%\)
🔑 Quick Example
A can do a job in 4 hours, B in 6 hours. Working together, how long?
\(\frac{1}{T}=\frac{1}{4}+\frac{1}{6}=\frac{5}{12}\), so \(T=\frac{12}{5}=2.4\) hours
Answer: 2.4 hours
Logarithms & Exponents
TOPIC 07 · LOGS & EXPONENTS
\(\log_b(xy) = \log_b x + \log_b y\)
\(\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y\)
\(\log_b(x^n) = n\log_b x\)
Change of base: \(\log_b x = \dfrac{\ln x}{\ln b}\)
\(\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y\)
\(\log_b(x^n) = n\log_b x\)
Change of base: \(\log_b x = \dfrac{\ln x}{\ln b}\)
📌 Must Memorize
- \(\log_b b = 1\), \(\log_b 1 = 0\)
- \(b^{\log_b x} = x\), \(\log_b b^x = x\)
- \(a^m \cdot a^n = a^{m+n}\), \((a^m)^n = a^{mn}\)
- \(a^{-n} = \dfrac{1}{a^n}\)
🔑 Quick Example
Simplify: \(\log_2 8 + \log_2 4\)
\(= 3 + 2 = 5\) (since \(\log_2 8=3\), \(\log_2 4=2\))
Answer: 5
Practice Problems
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