Algebra Cheat Sheet ... - Pauls Online Math Notes

Algebra Cheat SheetBasic Properties & FactsArithmetic OperationsProperties of Inequalities⎛b⎞ab If a< b thena+ c< b+ c and a− c< b−cab+ ac= ab ( + c)a⎜⎟=⎝ c ⎠ ca bIf a< b and c> 0 then ac< bc and bc and >c bc ⎛b⎞ bc c⎜ ⎟⎝c⎠Properties of Absolute Valuea c ad + bc a c ad −bc⎧aif a≥0+ = − =a =⎨b d bd b d bd⎩ − a if a < 0a−b b− a a+b a b a ≥0− a = a= = +c−d d −c c c ca aab = ab=⎛a⎞b bab+ ac⎜ ⎟b ad= b+ c, a ≠ 0⎝ ⎠=a+ b ≤ a + b Triangle Inequalitya⎛ c ⎞ bc⎜ ⎟⎝d⎠Distance FormulaExponent PropertiesIf Pn1= ( x1,y1)and P2 = ( x2,y2)are twon m n+ m a n−m1aa = a = a = points the distance between them ismm−na an( )mnm0a a a a( ab)a−n= = 1, ≠0n n n ⎛a⎞a= ab⎜ ⎟ =⎝b⎠b1 1 n= = an−naa−n n nnnmnnnn1m( ) ( ) 1mn⎛a⎞ ⎛b⎞b⎜ ⎟ = ⎜ ⎟ = a = a = a⎝b⎠ ⎝a⎠aProperties of Radicals1n nn n nm n nm nnna = a ab = a baa = a =bna = a,ifn is oddna = a ,if n is evennnab( , ) = ( − ) + ( − )2 2d P P x x y y1 2 2 1 2 1Complex Numbersi = − i =− − a = i a a≥( a+ bi) + ( c+ di) = a+ c+ ( b+d)i( a+ bi) − ( c+ di) = a− c+ ( b−d)i( + )( + ) = − + ( + )2 2( a+ bi)( a− bi)= a + b( )21 1 , 0a bi c di ac bd ad bc ia+ bi = a + b2 2a+ bi = a−bi( )( )a+ bi a+ bi = a+biComplex ModulusComplex Conjugate2For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.© 2005 Paul Dawkins

Logarithms and Log PropertiesDefinitionyy = log x is equivalent to x=bbExample3log 125= 3 because 5 = 1255Special Logarithmslnx=log x natural logelogx=log10x common logwhere e= 2.718281828KFactoring Formulas2 2x − a = x+ a x−a( )( )( )+ + = +2 2x 2ax a x a22 ( )( ) ( )( )33 3 ( )− + = −2 2x ax a x a2x + a+ b x+ ab= x+ a x+b3 2 2 3x + ax + ax+ a = x+a2( )3 2 2 3x − 3ax + 3ax− a = x−a( )( )( )( )( )( )x + a = x+ a x − ax+a3 3 2 2x − a = x− a x + ax+a3 3 2 2x − a = x − a x + a2n 2n n n n nIf n is odd then,x − a = x− a x + ax + L+ax−1 −2 −1( )( )n n n n nn+ ann−1 n−2 2 n−3 n−1( x a)( x ax ax L a )= + − + − +Solve22x−6x− 10=03Logarithm Propertieslog b= 1 log 1=0loglogbxlogbxbb = x b = xbr( )x = rlogx( )Factoring and Solvinglog xy = log x+log yb b b⎛ x⎞ log ⎜ ⎟ = log x − log⎝ y⎠b b bThe domain of log bx is x> 0Quadratic Formula2Solve ax + bx+ c= 0, a≠ 0If bIf bIf b222bb2− b±b −4acx=2a− 4ac> 0 - Two real unequal solns.− 4ac= 0 - Repeated real solution.− 4ac< 0 - Two complex solutions.Square Root Property2If x = p then x=±pAbsolute Value Equations/InequalitiesIf b is a positive numberp = b ⇒ p=− b or p = bp < b ⇒ − b< p b ⇒ p bCompleting the Square(4) Factor the left side2x(1) Divide by the coefficient of the2x −3x− 5=0(2) Move the constant to the other side.2x − 3x=5(3) Take half the coefficient of x, squareit and add it to both sides2 2x 2 3 3 9 29− 3 x + ⎛ ⎜− ⎞ ⎟ = 5+ ⎛ ⎜− ⎞⎟ = 5+ =⎝ 2⎠ ⎝ 2⎠4 4⎛ 3⎞29⎜x− ⎟ =⎝ 2⎠4(5) Use Square Root Property3 29 29x− =± =±2 4 2(6) Solve for x2y3 29x= ±2 2For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.© 2005 Paul Dawkins