📐 Algebra
Algebra Complete Cheatsheet
Linear equations, quadratics, polynomials, functions and sequences — complete algebra reference.
01Linear Equations▼
Slope
m=(y2-y1)/(x2-x1)
Rise over run.
Slope-intercept
y=mx+b
m=slope, b=y-intercept.
Standard
Ax+By=C
Integers, A≥0.
Midpoint
M=((x1+x2)/2,(y1+y2)/2)
ALGEBRASolving systems
# Substitution x+y=10, x-y=2 From 1st: x=10-y. Sub: 10-y-y=2. y=4, x=6 # Elimination 2x+3y=12 4x-3y=6 Add: 6x=18, x=3, y=2
02Quadratic Equations▼
Quadratic formula
x=(-b+-sqrt(b2-4ac))/2a
For ax2+bx+c=0.
Discriminant
D=b2-4ac
D>0: 2 roots, D=0: 1, D<0: none.
Vertex form
y=a(x-h)2+k
Vertex (h,k).
Factored
y=a(x-r1)(x-r2)
r1,r2 are roots.
ALGEBRAQuadratic examples
# Formula: 2x2+5x-3=0 a=2,b=5,c=-3. D=25+24=49 x=(-5+-7)/4. x=0.5 or x=-3 # Completing square: x2+6x+5=0 (x+3)2-9+5=0. (x+3)2=4. x=-3+-2
❓ Quiz
Discriminant of x2-6x+9=0?
b2-4ac=36-36=0. D=0 means one repeated root. Here x=3.
03Polynomials & Exponents▼
Power rules
am.an=am+n, (am)n=amn, a0=1, a-n=1/an
Difference of squares
a2-b2=(a+b)(a-b)
Perfect square
(a+b)2=a2+2ab+b2
Sum of cubes
a3+b3=(a+b)(a2-ab+b2)
Factor theorem
If f(a)=0 then (x-a) is a factor
Binomial
(a+b)n=sum C(n,k) an-k bk
ALGEBRAFactoring examples
x2-5x+6 = (x-2)(x-3) x3-8 = (x-2)(x2+2x+4) 2x3-8x = 2x(x2-4) = 2x(x+2)(x-2)
04Functions▼
Composition
(fog)(x)=f(g(x))
Inverse
Swap x,y then solve for y
Domain
Valid inputs. Watch: 1/0, sqrt(neg), log(neg)
Even
f(-x)=f(x), symmetric y-axis. eg x2
Odd
f(-x)=-f(x), symmetric origin. eg x3
Transformations
y=af(bx+c)+d: a=stretch, b=compress, c=shift, d=vert shift
💡
To find inverse: replace f(x) with y, swap x and y, solve for y. The graphs of f and f-1 are reflections over y=x.
05Sequences & Series▼
Arith nth term
an=a1+(n-1)d
d=common difference.
Arith sum
Sn=n/2(a1+an)
Sum of n terms.
Geo nth term
an=a1*r^(n-1)
r=common ratio.
Geo sum
Sn=a1(1-rn)/(1-r)
For r not equal 1.
Infinite geo
S=a1/(1-r)
Only when |r|<1.
ALGEBRASequences
# Arithmetic: 3,7,11... d=4 a10=3+(9*4)=39. S10=10/2*(3+39)=210 # Geometric: 2,6,18... r=3 a6=2*3^5=486. S6=2*(1-3^6)/(1-3)=728 # Infinite: 8,4,2... r=0.5 S=8/(1-0.5)=16