Formula Library

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Sets & Logic
De Morgan's Laws
(A∪B)′=A′∩B′(A∩B)′=A′∪B′(A\cup B)'=A'\cap B'\qquad (A\cap B)'=A'\cup B'
When to use: Simplifying complements of combined sets or logic statements.
Example: ¬(p∨q)≡¬p∧¬q\neg(p\vee q)\equiv\neg p\wedge\neg q
Easy
Sets & Logic
Inclusion–Exclusion
n(A∪B)=n(A)+n(B)−n(A∩B)n(A\cup B)=n(A)+n(B)-n(A\cap B)
When to use: Counting elements when two sets overlap.
Example: 18+14−6=2618+14-6=26 students take math or physics.
Easy
Algebra
Difference of Squares
a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
When to use: Fast factoring when both terms are perfect squares.
Example: x2−9=(x−3)(x+3)x^2-9=(x-3)(x+3)
Easy
Algebra
Vieta's Formulas
r1+r2=−bar1r2=car_1+r_2=-\frac{b}{a}\qquad r_1r_2=\frac{c}{a}
When to use: Getting root sums and products without solving the quadratic.
Example: 2x2−7x+32x^2-7x+3: sum =72=\frac{7}{2}, product =32=\frac{3}{2}
Medium
Quadratic
Quadratic Formula
x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
When to use: Solving any ax2+bx+c=0ax^2+bx+c=0, factorable or not.
Example: x2−5x+6=0⇒x=2, 3x^2-5x+6=0\Rightarrow x=2,\,3
Medium
Quadratic
Discriminant
Δ=b2−4ac\Delta=b^2-4ac
When to use: Predicting root type: Δ>0\Delta>0 two real, Δ=0\Delta=0 one repeated, Δ<0\Delta<0 none real.
Example: x2+x+1x^2+x+1: Δ=−3\Delta=-3 no real roots
Easy
Exponents & Logs
Laws of Exponents
aman=am+n(am)n=amna^m a^n=a^{m+n}\qquad (a^m)^n=a^{mn}
When to use: Combining powers with the same base.
Example: 23⋅24=27=1282^3\cdot 2^4=2^7=128
Easy
Exponents & Logs
Change of Base
log⁡bx=ln⁡xln⁡b\log_b x=\frac{\ln x}{\ln b}
When to use: Evaluating logs with any base on a calculator.
Example: log⁡210=ln⁡10ln⁡2≈3.32\log_2 10=\frac{\ln 10}{\ln 2}\approx 3.32
Medium
Functions
Composite Function
(f∘g)(x)=f(g(x))(f\circ g)(x)=f(g(x))
When to use: Chaining functions; setting up the chain rule.
Example: f(x)=2x+1,  g(x)=x2f(x)=2x+1,\;g(x)=x^2: (f∘g)(3)=19(f\circ g)(3)=19
Medium
Functions
Average Rate of Change
f(b)−f(a)b−a\frac{f(b)-f(a)}{b-a}
When to use: Slope of the secant line between two points on a graph.
Example: f(x)=x2f(x)=x^2 from 1 to 3: 9−12=4\frac{9-1}{2}=4
Medium
Geometry
Distance & Midpoint
d=(x2−x1)2+(y2−y1)2M=(x1+x22,y1+y22)d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\qquad M=\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)
When to use: Length and midpoint between two points in the plane.
Example: (0,0)(0,0) to (3,4)(3,4): d=5d=5
Easy
Geometry
Circle Equation
(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2
When to use: Reading center (h,k)(h,k) and radius rr; signs flip inside the brackets.
Example: (x−4)2+(y+2)2=25(x-4)^2+(y+2)^2=25: center (4,−2)(4,-2), r=5r=5
Easy
Trigonometry
Pythagorean Identity
sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1
When to use: Converting between sin and cos; simplifying identities.
Example: sin⁡θ=35⇒cos⁡θ=±45\sin\theta=\frac{3}{5}\Rightarrow\cos\theta=\pm\frac{4}{5}
Easy
Trigonometry
Law of Cosines
c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C
When to use: Solving triangles with two sides and the included angle.
Example: a=3,b=4,C=90°a=3,b=4,C=90°: c=5c=5
Medium
Sequences
Arithmetic nth Term & Sum
an=a1+(n−1)dSn=n2(a1+an)a_n=a_1+(n-1)d\qquad S_n=\frac{n}{2}(a_1+a_n)
When to use: Any term or partial sum with constant difference dd.
Example: 2,5,8,…2,5,8,\dots: a10=2+9⋅3=29a_{10}=2+9\cdot 3=29
Easy
Sequences
Geometric Series
an=a1rn−1S∞=a11−r  (∣r∣<1)a_n=a_1r^{n-1}\qquad S_\infty=\frac{a_1}{1-r}\;(|r|<1)
When to use: Terms and infinite sums with common ratio rr.
Example: a1=3,r=12a_1=3,r=\frac{1}{2}: S∞=6S_\infty=6
Medium
Probability
Addition Rule
P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)
When to use: Probability that at least one of two events occurs.
Example: 0.4+0.5−0.2=0.70.4+0.5-0.2=0.7
Easy
Probability
Permutations & Combinations
nPr=n!(n−r)!nCr=n!r!(n−r)!{}_nP_r=\frac{n!}{(n-r)!}\qquad {}_nC_r=\frac{n!}{r!(n-r)!}
When to use: Counting arrangements (order matters) vs selections (order doesn't).
Example: 5C2=10_5C_2=10 ways to pick 2 of 5
Medium
Statistics
Mean & Weighted Mean
xˉ=∑xinxˉ=∑fixi∑fi\bar{x}=\frac{\sum x_i}{n}\qquad \bar{x}=\frac{\sum f_ix_i}{\sum f_i}
When to use: The balancing point of a data set, plain or frequency-weighted.
Example: (2+4+8+10)/4=6(2+4+8+10)/4=6
Easy
Statistics
Z-score
z=x−μσz=\frac{x-\mu}{\sigma}
When to use: How many standard deviations a value sits from the mean.
Example: z=2,μ=50,σ=5⇒x=60z=2,\mu=50,\sigma=5\Rightarrow x=60
Medium
Matrices
2×2 Determinant
det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc
When to use: Invertibility test; nonzero det⁡\det means an inverse exists.
Example: rows (2  1),(3  4)(2\;1),(3\;4): det⁡=5\det=5
Easy
Matrices
2×2 Inverse
A−1=1ad−bc(d−b−ca)A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}
When to use: Solving AX=BAX=B when det⁡A≠0\det A\ne 0.
Example: Swap a,da,d; negate b,cb,c; divide by det⁡\det.
Hard
Complex Numbers
Modulus & Conjugate
∣a+bi∣=a2+b2a+bi‾=a−bi|a+bi|=\sqrt{a^2+b^2}\qquad \overline{a+bi}=a-bi
When to use: Distance from the origin; rationalizing complex denominators.
Example: ∣3+4i∣=5|3+4i|=5, (2+i)(2−i)=5(2+i)(2-i)=5
Easy
Complex Numbers
Powers of i
i2=−1in+4=ini^2=-1\qquad i^{n+4}=i^n
When to use: Reducing any power of ii via the 4-cycle i,−1,−i,1i,-1,-i,1.
Example: i15=i3=−ii^{15}=i^3=-i
Medium
Vectors
Dot Product
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=a1b1+a2b2\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_1b_1+a_2b_2
When to use: Angle between vectors; perpendicular exactly when it equals 0.
Example: ⟨1,2⟩⋅⟨3,4⟩=11\langle 1,2\rangle\cdot\langle 3,4\rangle=11
Medium
Vectors
Magnitude & Unit Vector
∣v⃗∣=x2+y2v^=v⃗∣v⃗∣|\vec{v}|=\sqrt{x^2+y^2}\qquad \hat{v}=\frac{\vec{v}}{|\vec{v}|}
When to use: Length of a vector and the direction-only version of it.
Example: ∣⟨3,4⟩∣=5|\langle 3,4\rangle|=5
Easy
Limits
Standard Trig Limits
lim⁡x→0sin⁡xx=1lim⁡x→0tan⁡xx=1\lim_{x\to 0}\frac{\sin x}{x}=1\qquad \lim_{x\to 0}\frac{\tan x}{x}=1
When to use: Evaluating trig limits; deriving (sin⁡x)′=cos⁡x(\sin x)'=\cos x.
Example: lim⁡sin⁡3xx=3\lim\frac{\sin 3x}{x}=3
Medium
Limits
Rational Limits at Infinity
deg⁡P<deg⁡Q⇒0deg⁡P=deg⁡Q⇒aleadblead\deg P<\deg Q\Rightarrow 0\qquad \deg P=\deg Q\Rightarrow\frac{a_{\text{lead}}}{b_{\text{lead}}}
When to use: End behavior of P(x)/Q(x)P(x)/Q(x) by comparing degrees.
Example: 2x2+15x2−3→25\frac{2x^2+1}{5x^2-3}\to\frac{2}{5}
Hard
Derivatives
Power Rule
ddxxn=nxn−1\frac{d}{dx}x^n=nx^{n-1}
When to use: Differentiating polynomials term by term.
Example: (x7)′=7x6(x^7)'=7x^6
Easy
Derivatives
Product & Chain Rules
(uv)′=u′v+uv′[f(g(x))]′=f′(g(x)) g′(x)(uv)'=u'v+uv'\qquad [f(g(x))]'=f'(g(x))\,g'(x)
When to use: Differentiating products and nested functions.
Example: (xsin⁡x)′=sin⁡x+xcos⁡x(x\sin x)'=\sin x+x\cos x
Medium
Integration
Power Rule (Integrals)
∫xn dx=xn+1n+1+C  (n≠−1)∫1x dx=ln⁡∣x∣+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\;(n\ne -1)\qquad \int\frac{1}{x}\,dx=\ln|x|+C
When to use: Antiderivatives of polynomial terms, plus the n=−1n=-1 special case.
Example: ∫x2 dx=x33+C\int x^2\,dx=\frac{x^3}{3}+C
Easy
Integration
Fundamental Theorem
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a)
When to use: Evaluating definite integrals from any antiderivative FF.
Example: ∫012x dx=12−02=1\int_0^1 2x\,dx=1^2-0^2=1
Medium