Math formula sheet

Every formula with a worked example, from algebra to calculus.

Algebra formulas

Quadratic formula

x = (βˆ’b Β± √(b2 βˆ’ 4ac)) / 2a

Examplex2 βˆ’ 5x + 6 = 0: x = (5 Β± √1) / 2, so x = 3 or x = 2

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Slope

m = (y2 βˆ’ y1) / (x2 βˆ’ x1)

Example(1, 2) and (4, 11): m = 9 / 3 = 3

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Point-slope form

y βˆ’ y1 = m(x βˆ’ x1)

ExampleSlope 3 through (1, 2): y βˆ’ 2 = 3(x βˆ’ 1), so y = 3x βˆ’ 1

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Exponent rules

am Β· an = am+n
(am)n = amn
aβˆ’n = 1 / an

Example23 Β· 24 = 27 = 128

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Logarithms

logb x = y means by = x

Examplelog2 32 = 5 because 25 = 32

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Log rules

log(ab) = log a + log b
log(a/b) = log a βˆ’ log b
log(an) = n log a

Examplelog2 8 + log2 4 = log2 32 = 5

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Slope-intercept form

y = mx + b

ExampleSlope 2, intercept βˆ’3: at x = 4, y = 2(4) βˆ’ 3 = 5

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Arithmetic sequence

an = a1 + (n βˆ’ 1)d

Examplea1 = 3, d = 4: a10 = 3 + 9 Γ— 4 = 39

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Geometric sequence

an = a1 Β· rnβˆ’1

Examplea1 = 2, r = 3: aβ‚… = 2 Γ— 34 = 162

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Numbers and money formulas

Percent change

(new βˆ’ old) / old Γ— 100%

Example40 to 50: 10 / 40 = 0.25, a 25% increase

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Compound interest

A = P(1 + r/n)nt

Example$1,000 at 5% compounded yearly for 3 years: 1,000 Γ— 1.053 = $1,157.63

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Simple interest

I = Prt

Example$2,000 at 4% for 3 years: I = 2,000 Γ— 0.04 Γ— 3 = $240

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Exponential growth and decay

A = A0(1 + r)t

Example500 growing 10% a year for 2 years: 500 Γ— 1.12 = 605

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Geometry and trig formulas

Pythagorean theorem

a2 + b2 = c2

ExampleLegs 5 and 12: c = √(25 + 144) = √169 = 13

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Distance

d = √((x2 βˆ’ x1)2 + (y2 βˆ’ y1)2)

Example(1, 2) to (4, 6): √(9 + 16) = 5

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Midpoint

((x1 + x2)/2, (y1 + y2)/2)

Example(2, 3) and (8, 7): (10/2, 10/2) = (5, 5)

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Triangle area

A = Β½bh

ExampleBase 10, height 6: A = Β½ Γ— 10 Γ— 6 = 30

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Trapezoid area

A = Β½(b1 + b2)h

ExampleBases 6 and 10, height 4: Β½ Γ— 16 Γ— 4 = 32

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Circle

A = Ο€r2
C = 2Ο€r

Exampler = 3: A = 9Ο€ β‰ˆ 28.27, C = 6Ο€ β‰ˆ 18.85

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Cylinder volume

V = Ο€r2h

Exampler = 2, h = 5: V = 20Ο€ β‰ˆ 62.83

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Sphere volume

V = (4/3)Ο€r3

Exampler = 3: V = 36Ο€ β‰ˆ 113.10

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Law of cosines

c2 = a2 + b2 βˆ’ 2ab cos C

Examplea = 5, b = 7, C = 60Β°: 25 + 49 βˆ’ 35 = 39, so c = √39 β‰ˆ 6.24

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SOH CAH TOA

sin ΞΈ = opp / hyp
cos ΞΈ = adj / hyp
tan ΞΈ = opp / adj

ExampleOpposite 3, hypotenuse 5: sin ΞΈ = 0.6, so ΞΈ β‰ˆ 36.87Β°

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Statistics formulas

Mean and median

mean = sum / count
median = middle value, sorted

Example2, 4, 6, 8: mean = 20 / 4 = 5, median = (4 + 6) / 2 = 5

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Combinations

nCr = n! / (r!(n βˆ’ r)!)

ExampleChoose 3 of 5: 5! / (3! Β· 2!) = 120 / 12 = 10

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Permutations

nPr = n! / (n βˆ’ r)!

Example1st, 2nd and 3rd from 10 runners: 10 Γ— 9 Γ— 8 = 720

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Probability

P(A) = favorable outcomes / total outcomes
P(A and B) = P(A) Β· P(B) if independent

ExampleTwo coins both heads: Β½ Γ— Β½ = ΒΌ

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Z-score

z = (x βˆ’ ΞΌ) / Οƒ

ExampleScore 85, mean 70, SD 10: z = 15 / 10 = 1.5

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Calculus formulas

Power rule

d/dx xn = nxnβˆ’1

Exampled/dx 3x4 = 12x3

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Chain rule

d/dx f(g(x)) = fβ€²(g(x)) Β· gβ€²(x)

Exampled/dx (2x + 1)3 = 3(2x + 1)2 Β· 2 = 6(2x + 1)2

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Product rule

(fg)β€² = fβ€²g + fgβ€²

Exampled/dx x2(x + 1) = 2x(x + 1) + x2 = 3x2 + 2x

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Quotient rule

(f/g)β€² = (fβ€²g βˆ’ fgβ€²) / g2

Exampled/dx x/(x + 1) = ((x + 1) βˆ’ x)/(x + 1)2 = 1/(x + 1)2

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Integral power rule

∫ xn dx = xn+1 / (n + 1) + C, n β‰  βˆ’1

Example∫02 3x2 dx = [x3]02 = 8

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