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Algebra Cheat Sheet Basic Properties & Facts Arithmetic Operations ab + ac = a ( b + c ) æaö ç ÷ a èbø = c bc a c ad + bc + = b d bd a-b b-a = c-d d -c ab + ac = b + c, a ¹ 0 a Exponent Properties a na m = a n+ m an 1 = a n -m = m -n am a a 0 = 1, a ¹ 0 a æaö ç ÷ = n b èbø 1 n =a a-n n n n Logarithms and Log Properties Definition y = log b x is equivalent to x = b y Example log 5 125 = 3 because 53 = 125 Special Logarithms ln x = log e x natural log log x = log 10 x common log where e = 2.718281828K Factoring Formulas x 2 - a 2 = ( x + a )( x - a ) x 2 + 2ax + a 2 = ( x + a ) x 2 - 2 ax + a 2 = ( x - a ) 2 2 Logarithm Properties log b b = 1 log b 1 = 0 log b b x = x log b ( x r æ b ö ab aç ÷ = ècø c a ac = æbö b çc÷ è ø a c ad - bc - = b d bd a+b a b = + c c c æaö ç b ÷ ad è ø= æ c ö bc çd ÷ è ø Properties of Inequalities If a < b then a + c < b + c and a - c < b - c a b < c c a b If a < b and c < 0 then ac > bc and > c c If a < b and c > 0 then ac < bc and Properties of Absolute Value if a ³ 0 ìa a =í if a < 0 î-a a ³0 -a = a ab = a b a +b £ a + b a a = b b Triangle Inequality b logb x = x b ) = r log x log b ( xy ) = log b x + logb y æxö log b ç ÷ = log b x - logb y è yø The domain of log b x is x > 0 Factoring and Solving Quadratic Formula Solve ax 2 + bx + c = 0 , a ¹ 0 -b ± b 2 - 4 ac 2a If b 2 - 4ac > 0 - Two real unequal solns. If b 2 - 4ac = 0 - Repeated real solution. If b 2 - 4ac < 0 - Two complex solutions. x= Square Root Property If x2 = p then x = ± p Absolute Value Equations/Inequalities If b is a positive number p =b Þ p = - b or p = b p b Þ Þ -b < p < b p < - b or p>b x 2 + ( a + b ) x + ab = ( x + a )( x + b ) x3 + 3ax 2 + 3a 2 x + a 3 = ( x + a ) x3 - 3ax2 + 3a 2 x - a 3 = ( x - a ) 3 3 Distance Formula If P = ( x1, y1 ) and P2 = ( x2 , y2 ) are two 1 points the distance between them is d ( P , P2 ) = 1 (a ) n m x3 + a 3 = ( x + a ) ( x2 - ax + a 2 ) x3 - a 3 = ( x - a ) ( x2 + ax + a 2 ) x 2 n - a 2 n = ( x n - a n )( x n + a n ) If n is odd then, x n - a n = ( x - a ) ( x n-1 + ax n -2 + L + a n -1 ) xn + an = ( x + a)( x n -1 = a nm =a b n n ( x2 - x1 ) 2 + ( y2 - y1 ) 2 ( ab ) a -n n Complex Numbers i = -1 n 1 m 1 = n a -n i = -1 2 -a = i a, a ³ 0 æaö ç ÷ èbø bn æbö =ç ÷ = n a è aø a = a n m ( ) = (a ) 1 m n Properties of Radicals n ( a + bi ) + ( c + di ) = a + c + ( b + d ) i ( a + bi ) - ( c + di ) = a - c + ( b - d ) i ( a + bi )( c + di ) = ac - bd + ( ad + bc ) i ( a + bi )( a - bi ) = a 2 + b2 a + bi = a + b 2 2 - ax n -2 +a x 2 n -3 -L + a n -1 ) 3ö 29 æ çx- ÷ = 2ø 4 è (5) Use Square Root Property 3 29 29 x- =± =± 2 4 2 (6) Solve for x 3 29 x= ± 2 2 2 Solve 2 x 2 - 6 x - 10 = 0 Completing the Square (4) Factor the left side 2 a =a 1 n n ab = a b n n Complex Modulus m n a = nm a n a na = b nb n n a n = a , if n is odd a n = a , if n is even ( a + bi ) = a - bi Complex Conjugate 2 ( a + bi )( a + bi ) = a + bi (1) Divide by the coefficient of the x x 2 - 3x - 5 = 0 (2) Move the constant to the other side. x 2 - 3x = 5 (3) Take half the coefficient of x, square it and add it to both sides 9 29 æ 3ö æ 3ö x 2 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 4 4 è 2ø è 2ø 2 2 For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins Functions and Graphs Constant Function y = a or f ( x) = a Graph is a horizontal line passing through the point ( 0, a ) . Line/Linear Function y = mx + b or f ( x ) = mx + b Graph is a line with point ( 0,b ) and slope m. Slope Slope of the line containing the two points ( x1 , y1 ) and ( x2 , y2 ) is y2 - y1 rise = x2 - x1 run Slope – intercept form The equation of the line with slope m and y-intercept ( 0,b ) is y = mx + b Point – Slope form The equation of the line with slope m and passing through the point ( x1 , y1 ) is m= y = y1 + m ( x - x1 ) Parabola/Quadratic Function y = a ( x - h) + k 2 Common Algebraic Errors Error 2 2 ¹ 0 and ¹ 2 0 0 -32 ¹ 9 Reason/Correct/Justification/Example Division by zero is undefined! -32 = -9 , Parabola/Quadratic Function x = ay2 + by + c g ( y ) = ay2 + by + c The graph is a parabola that opens right if a > 0 or left if a < 0 and has a vertex æ æ b ö b ö at ç g ç - ÷ , - ÷ . è è 2a ø 2a ø Circle 2 2 ( x - h) + ( y - k ) = r2 Graph is a circle with radius r and center ( h, k ) . Ellipse =1 a2 b2 Graph is an ellipse with center ( h, k ) with vertices a units right/left from the center and vertices b units up/down from the center. Hyperbola ( -3) 2 = 9 Watch parenthesis! (x 2 3 ) ¹ x5 (x 2 3 ) = x2 x2 x 2 = x6 a a a ¹ + b+c b c 1 ¹ x -2 + x-3 x2 + x3 a + bx ¹ 1 + bx a - a ( x - 1) ¹ - ax - a ( x - h) 2 ( y - k) + 2 1 1 1 1 = ¹ + =2 2 1+ 1 1 1 A more complex version of the previous error. a + bx a bx bx = + =1+ a a a a Beware of incorrect canceling! - a ( x - 1) = - ax + a Make sure you distribute the “-“! ( x + a) 2 ¹ x2 + a 2 x+ a n ( x + a) 2 = ( x + a )( x + a ) = x 2 + 2 ax + a 2 x2 + a2 ¹ x + a x+a ¹ ( x + a) n ¹ x n + a n and 2 2 x+a ¹ n x+n a 5 = 25 = 3 2 + 4 2 ¹ 3 2 + 4 2 = 3 + 4 = 7 See previous error. More general versions of previous three errors. 2 ( x + 1) = 2 ( x2 + 2 x + 1) = 2 x 2 + 4 x + 2 2 2 ( x - h) 2 2 f ( x) = a ( x - h) + k 2 The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at ( h, k ) . Parabola/Quadratic Function y = ax 2 + bx + c f ( x ) = ax 2 + bx + c The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex æ b æ b öö at ç - , f ç - ÷ ÷ . è 2a è 2a ø ø a b Graph is a hyperbola that opens left and right, has a center at ( h, k ) , vertices a units left/right of center and asymptotes b that pass through center with slope ± . a Hyperbola - ( y - k) 2 2 =1 2 ( x + 1) ¹ ( 2 x + 2 ) ( 2x + 2) 2 ¹ 2 ( x + 1) 2 = 4 x2 + 8x + 4 Square first then distribute! See the previous example. You can not factor out a constant if there is a power on the parethesis! - x2 + a2 = ( - x2 + a2 )2 Now see the previous error. æaö ç1÷ a æ a ö æ c ö ac = è ø = ç ÷ç ÷ = æ b ö æ b ö è 1 øè b ø b ç ÷ ç ÷ ècø ècø æaö æaö ç ÷ ç ÷ è b ø = è b ø = æ a öæ 1 ö = a ç ÷ç ÷ c æ c ö è b ø è c ø bc ç ÷ 1ø è 1 ( 2x + 2) - x2 + a2 ¹ - x2 + a2 a ab ¹ æbö c çc÷ è ø æaö ç ÷ ac èbø ¹ c b ( x - h) = 1 b2 a2 Graph is a hyperbola that opens up and down, has a center at ( h, k ) , vertices b units up/down from the center and asymptotes that pass through center with b slope ± . a 2 2 (y -k) For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
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