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Algebra cheat sheet

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  • 1. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul DawkinsAlgebra Cheat SheetBasic Properties & FactsArithmetic Operations( ), 0b abab ac a b c ac caa a acbbc bc bca c ad bc a c ad bcb d bd b d bda b b a a b a bc d d c c c caab ac adbb c aca bcd + = + =      = =   + −+ = − =− − += = +− −  +  = + ≠ =   Exponent Properties( )( )( ) ( )11011, 01 1nmm mnn m n m n mm m nmn nmn nn n nnn nn nn n n nnnaa a a aa aa a a aa aab a bb ba aa aa b ba a ab a a+ −−−−−= = == = ≠ = =  = =   = = = =      Properties of Radicals1, if is odd, if is evennn n n nnm n nm nnn nn na a ab a ba aa ab ba a na a n= == ===Properties of InequalitiesIf then andIf and 0 then andIf and 0 then anda b a c b c a c b ca ba b c ac bcc ca ba b c ac bcc c< + < + − < −< > < << < > >Properties of Absolute Valueif 0if 0a aaa a≥= − <0Triangle Inequalitya a aaaab a bb ba b a b≥ − == =+ ≤ +Distance FormulaIf ( )1 1 1,P x y= and ( )2 2 2,P x y= are twopoints the distance between them is( ) ( ) ( )2 21 2 2 1 2 1,d P P x x y y= − + −Complex Numbers( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )( )( )( )( )22 22 221 1 , 0Complex ModulusComplex Conjugatei i a i a aa bi c di a c b d ia bi c di a c b d ia bi c di ac bd ad bc ia bi a bi a ba bi a ba bi a bia bi a bi a bi= − = − − = ≥+ + + = + + ++ − + = − + −+ + = − + ++ − = ++ = ++ = −+ + = +
  • 2. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul DawkinsLogarithms and Log PropertiesDefinitionlog is equivalent to yby x x b= =Example35log 125 3 because 5 125= =Special Logarithms10ln log natural loglog log common logex xx x==where 2.718281828e = KLogarithm Properties( )( )loglog 1 log 1 0loglog loglog log loglog log logbb bxxbrb bb b bb b bbb x b xx r xxy x yxx yy= == === + = −  The domain of logb x is 0x >Factoring and SolvingFactoring Formulas( )( )( )( )( ) ( )( )2 222 222 2222x a x a x ax ax a x ax ax a x ax a b x ab x a x b− = + −+ + = +− + = −+ + + = + +( )( )( )( )( )( )33 2 2 333 2 2 33 3 2 23 3 2 23 33 3x ax a x a x ax ax a x a x ax a x a x ax ax a x a x ax a+ + + = +− + − = −+ = + − +− = − + +( )( )2 2n n n n n nx a x a x a− = − +If n is odd then,( )( )( )( )1 2 11 2 2 3 1n n n n nn nn n n nx a x a x ax ax ax a x ax a x a− − −− − − −− = − + + ++= + − + − +LLQuadratic FormulaSolve 20ax bx c+ + = , 0a ≠242b b acxa− ± −=If 24 0b ac− > - Two real unequal solns.If 24 0b ac− = - Repeated real solution.If 24 0b ac− < - Two complex solutions.Square Root PropertyIf 2x p= then x p= ±Absolute Value Equations/InequalitiesIf b is a positive numberororp b p b p bp b b p bp b p b p b= ⇒ = − =< ⇒ − < <> ⇒ < − >Completing the SquareSolve 22 6 10 0x x− − =(1) Divide by the coefficient of the 2x23 5 0x x− − =(2) Move the constant to the other side.23 5x x− =(3) Take half the coefficient of x, squareit and add it to both sides2 22 3 3 9 293 5 52 2 4 4x x   − + − = + − = + =      (4) Factor the left side23 292 4x − =  (5) Use Square Root Property3 29 292 4 2x − = ± = ±(6) Solve for x3 292 2x = ±
  • 3. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul DawkinsFunctions and GraphsConstant Function( )ory a f x a= =Graph is a horizontal line passingthrough the point ( )0,a .Line/Linear Function( )ory mx b f x mx b= + = +Graph is a line with point ( )0,b andslope m.SlopeSlope of the line containing the twopoints ( )1 1,x y and ( )2 2,x y is2 12 1riseruny ymx x−= =−Slope – intercept formThe equation of the line with slope mand y-intercept ( )0,b isy mx b= +Point – Slope formThe equation of the line with slope mand passing through the point ( )1 1,x y is( )1 1y y m x x= + −Parabola/Quadratic Function( ) ( ) ( )2 2y a x h k f x a x h k= − + = − +The graph is a parabola that opens up if0a > or down if 0a < and has a vertexat ( ),h k .Parabola/Quadratic Function( )2 2y ax bx c f x ax bx c= + + = + +The graph is a parabola that opens up if0a > or down if 0a < and has a vertexat ,2 2b bfa a  − −    .Parabola/Quadratic Function( )2 2x ay by c g y ay by c= + + = + +The graph is a parabola that opens rightif 0a > or left if 0a < and has a vertexat ,2 2b bga a  − −    .Circle( ) ( )2 2 2x h y k r− + − =Graph is a circle with radius r and center( ),h k .Ellipse( ) ( )2 22 21x h y ka b− −+ =Graph is an ellipse with center ( ),h kwith vertices a units right/left from thecenter and vertices b units up/down fromthe center.Hyperbola( ) ( )2 22 21x h y ka b− −− =Graph is a hyperbola that opens left andright, has a center at ( ),h k , vertices aunits left/right of center and asymptotesthat pass through center with slopeba± .Hyperbola( ) ( )2 22 21y k x hb a− −− =Graph is a hyperbola that opens up anddown, has a center at ( ),h k , vertices bunits up/down from the center andasymptotes that pass through center withslopeba± .
  • 4. For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul DawkinsCommon Algebraic ErrorsError Reason/Correct/Justification/Example200≠ and220≠ Division by zero is undefined!23 9− ≠23 9− = − , ( )23 9− = Watch parenthesis!( )32 5x x≠ ( )32 2 2 2 6x x x x x= =a a ab c b c≠ ++1 1 1 122 1 1 1 1= ≠ + =+2 32 31x xx x− −≠ ++A more complex version of the previouserror.a bxa+1 bx≠ +1a bx a bx bxa a a a+= + = +Beware of incorrect canceling!( )1a x ax a− − ≠ − −( )1a x ax a− − = − +Make sure you distribute the “-“!( )2 2 2x a x a+ ≠ + ( ) ( )( )2 2 22x a x a x a x ax a+ = + + = + +2 2x a x a+ ≠ + 2 2 2 25 25 3 4 3 4 3 4 7= = + ≠ + = + =x a x a+ ≠ + See previous error.( )n n nx a x a+ ≠ + and n n nx a x a+ ≠ +More general versions of previous threeerrors.( ) ( )2 22 1 2 2x x+ ≠ +( ) ( )2 2 22 1 2 2 1 2 4 2x x x x x+ = + + = + +( )2 22 2 4 8 4x x x+ = + +Square first then distribute!( ) ( )2 22 2 2 1x x+ ≠ +See the previous example. You can notfactor out a constant if there is a power onthe parethesis!2 2 2 2x a x a− + ≠ − + ( )12 2 2 2 2x a x a− + = − +Now see the previous error.a abb cc≠   11aa a c acb b b bc c     = = =              aacbc b    ≠11a aa ab bcc b c bc           = = =        