Every Formula
Instantly

Algebra Essentials

The most frequently referenced algebra formulas:

• Quadratic formula: x=(−b±√(b²−4ac))/2a — solves any ax²+bx+c=0
• Difference of squares: a²−b²=(a+b)(a−b)
• Perfect square trinomials: (a+b)²=a²+2ab+b², (a−b)²=a²−2ab+b²
• Slope-intercept form: y=mx+b (m=slope, b=y-intercept)
• Point-slope form: y−y₁=m(x−x₁)

The discriminant b²−4ac tells you about roots before solving: positive=two real roots, zero=one repeated root, negative=two complex roots.

Geometry — Area, Volume, and Pythagorean Theorem

• Circle: Area=πr², Circumference=2πr
• Triangle: Area=½bh; Heron's formula: √(s(s−a)(s−b)(s−c)) where s=(a+b+c)/2
• Pythagorean theorem: a²+b²=c² for right triangles
• Sphere: Volume=(4/3)πr³, Surface area=4πr²
• Cylinder: Volume=πr²h, Lateral surface=2πrh
• Cone: Volume=(1/3)πr²h

Over 370 distinct proofs of the Pythagorean theorem exist, including a geometric proof attributed to US President James Garfield.

Trigonometry — Functions and Key Identities

• SOH-CAH-TOA: sin=Opp/Hyp, cos=Adj/Hyp, tan=Opp/Adj
• Pythagorean identity: sin²θ+cos²θ=1
• Double angle: sin(2θ)=2sinθcosθ, cos(2θ)=cos²θ−sin²θ
• Sum formulas: sin(A+B)=sinAcosB+cosAsinB
• Law of cosines: c²=a²+b²−2ab·cos(C)
• Law of sines: a/sin(A)=b/sin(B)=c/sin(C)

Key values: sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2, cos(0°)=1, cos(90°)=0.

Calculus — Derivatives and Integrals

Most-used differentiation rules:

• Power rule: d/dx[xⁿ]=nxⁿ⁻¹
• Product rule: d/dx[f·g]=f'g+fg'
• Quotient rule: d/dx[f/g]=(f'g−fg')/g²
• Chain rule: d/dx[f(g(x))]=f'(g(x))·g'(x)

Common integrals: ∫xⁿdx=xⁿ⁺¹/(n+1)+C, ∫eˣdx=eˣ+C, ∫(1/x)dx=ln|x|+C, ∫sin(x)dx=−cos(x)+C, ∫cos(x)dx=sin(x)+C.

Statistics and Probability

• Mean: μ=Σxᵢ/n
• Standard deviation: σ=√(Σ(xᵢ−μ)²/n) for population; use n−1 for sample
• Z-score: z=(x−μ)/σ
• Combinations: C(n,r)=n!/(r!(n−r)!)
• Permutations: P(n,r)=n!/(n−r)!
• Bayes' theorem: P(A|B)=P(B|A)·P(A)/P(B)

The 68-95-99.7 rule: in a normal distribution, 68% of data falls within 1σ of the mean, 95% within 2σ, 99.7% within 3σ.

Physics — Classical Mechanics

• Newton's second law: F=ma
• Kinetic energy: KE=½mv²
• Gravitational PE: PE=mgh
• Projectile range: R=v₀²·sin(2θ)/g
• Ohm's law: V=IR
• Power: P=IV=I²R=V²/R
• Wave speed: v=fλ

g=9.81 m/s² at Earth's surface. Many problems use g=10 m/s² for easier mental arithmetic.

Euler's Identity and Beautiful Mathematics

Euler's identity — e^(iπ)+1=0 — connects the five most important constants in mathematics: e, i, π, 1, and 0. In a 1988 survey, physicists and mathematicians voted it the most beautiful equation in mathematics. It arises from Euler's formula e^(iθ)=cosθ+i·sinθ; substituting θ=π gives cos(π)=−1, sin(π)=0, so e^(iπ)=−1.

Financial Mathematics

• Simple interest: I=Prt
• Compound interest: A=P(1+r/n)^(nt)
• Continuous compounding: A=Pe^(rt)
• Rule of 72: years to double ≈ 72/r%
• Present value: PV=FV/(1+r)ⁿ
• Loan payment: PMT=P·[r(1+r)ⁿ]/[(1+r)ⁿ−1]

The Rule of 72 is one of the most useful mental math shortcuts in finance — doubling-time estimates without a calculator.