1 · Systems of Linear Equations
Solve using substitution or elimination. For 2×2 system \(ax+by=c\), \(dx+ey=f\).
Example: \(2x+3y=12,\; x-y=1\)
From 2nd: \(x=y+1\). Sub: \(2(y+1)+3y=12 \Rightarrow 5y=10 \Rightarrow y=2, x=3\).
2 · Quadratic Equations
x = (-b ± √(b²-4ac)) / 2a
Discriminant \(D=b^2-4ac\): \(D>0\) two real roots, \(D=0\) one root, \(D<0\) two complex roots.
Example: \(x^2-6x+5=0 \Rightarrow (x-1)(x-5)=0 \Rightarrow x=1,5\)
3 · Complex Numbers
i² = −1 | (a+bi)(c+di) = (ac−bd)+(ad+bc)i
Multiply by distributing; replace \(i^2=-1\). Modulus: \(|a+bi|=\sqrt{a^2+b^2}\).
Example: \((3+2i)(1-i)=3-3i+2i-2i^2=5-i\)
4 · Polynomial Division
f(x) = d(x)·q(x) + r
Synthetic division: when dividing by \((x-k)\), use \(k\) as the divisor. Remainder Theorem: \(r = f(k)\).
Example: \((x^3-2x^2+x-3)\div(x-2)=x^2+1\) remainder \(-1\)
5 · Factoring Special Forms
a³+b³ = (a+b)(a²−ab+b²)
a³−b³ = (a−b)(a²+ab+b²)
Example: \(8x^3+27=(2x+3)(4x^2-6x+9)\)
6 · Rational Expressions
Factor numerator and denominator completely, then cancel common factors (state excluded values).
Example: \(\dfrac{x^2-4}{x+2}=\dfrac{(x+2)(x-2)}{x+2}=x-2,\; x\neq-2\)
7 · Radical Equations
Isolate the radical, then square both sides. Always check for extraneous solutions.
Example: \(\sqrt{x+3}=x-3\Rightarrow x+3=(x-3)^2\Rightarrow x=6\) (check: \(x=1\) extraneous)
8 · Exponential & Logarithmic Equations
log_b(mn) = log_b m + log_b n
b^x = b^y ⟹ x = y
Example: \(2^{x+1}=32=2^5 \Rightarrow x+1=5 \Rightarrow x=4\)
9 · Sequences & Series
Arithmetic: a_n = a₁+(n-1)d | S_n = n(a₁+a_n)/2
Geometric: a_n = a₁·rⁿ⁻¹ | S_n = a₁(rⁿ-1)/(r-1)
Example: \(a_1=3, r=2 \Rightarrow S_6=3(2^6-1)/(2-1)=189\)
10 · Matrices & Determinants
det[a,b;c,d] = ad − bc
For 2×2 product \([A][B]\): row × column. Determinant determines invertibility (\(\det\neq 0\)).
11 · Conic Sections
Parabola: x²=4py (focus at (0,p))
Ellipse: x²/a²+y²/b²=1, c²=a²-b²
12 · Binomial Theorem & Combinations
(a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ
C(n,r) = n! / (r!(n-r)!)
Example: In \((x+2)^4\), coeff of \(x^2\) = \(\binom{4}{2}\cdot2^2=24\)