vedic mathematics

For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins Algebra Cheat Sheet Basic Properties & Facts Arithmetic Operations ( ) , 0 ab ac a b c a b ab c c a b a a ac c bc b b c a c ad bc a c ad bc b d bd b d bd a b b a a b a b c d d c c c c a ab ac b c a b ad a c bc d+ = + æ ö = ç ÷ è ø æ ö ç ÷ è ø= = æ ö ç ÷ è ø + - + = - = - - + = = + - - æ ö + çè ÷ø = + ¹ = æ ö ç ÷ è ø Exponent Properties ( ) ( ) ( ) ( )1 1 0 1 1, 0 1 1n m m m n n m n m n m m m n n m nm n n n n n n n n n n n n n n n n a a a a a a a a a a a ab a b a a b b a a a a a b b a a a b a a + - - - - - = = = = = ¹ = æ ö = ç ÷ è ø = = æ ö = æ ö = = = ç ÷ ç ÷ è ø è ø Properties of Radicals 1 , if is odd , if is even n n n n n n mn nm n n n n n n aa ab a b a a a a b b a a n a a n = = = = == Properties of Inequalities If then and If and 0 then and If and 0 then and a b a c b c a c b c a b c ac bc a b c c a b c ac bc a b c c < + <+ - < - < > < < < < > > Properties of Absolute Value if 0 if 0 a a a a aì ³ = í- < î0 Triangle Inequality a a aa a ab a b b b a b a b ³ - = = = + £ + Distance Formula If ( ) 1 1 1 P = x , y and ( ) 2 2 2 P = x , y are two points the distance between them is ( ) ( )2 ( )2 1 2 2 1 2 1 d P, P = x - x + y - y Complex Numbers ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) 2 2 2 2 2 2 1 1 , 0 Complex Modulus Complex Conjugate i i a i a a 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 b a bi a b a bi a bi a bi a bi a bi =- =- -= ³ + + + = + + + + - + = - + - + + = - + + + - = + + = + + = - + + = +For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins Logarithms and Log Properties Definition log is equivalent to y by= x x = b Example 3 5 log 125 = 3 because 5 =125 Special Logarithms 10 ln log natural log log log common log e x x x x == where e = 2.718281828K Logarithm Properties ( ) ( ) log log 1 log 1 0 log log log log log log log log log b b bx x b r b b b b b b b b bb x b x x r x xy x y x x y y= = = = == + æ öç ÷ = - è ø The domain of logb x is x > 0 Factoring and Solving Factoring Formulas ( )( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 22 2 2 222 x a x a x a x ax a x a x ax a x a x a b x ab x a x b - = + - + + = + - + = - + + + = + + ( ) ( ) ( )( ) ( )( )3 2 2 3 33 2 2 3 3 3 3 2 2 3 3 2 2 3 3 3 3 x ax a x a x a x ax a x a x a x a x a x ax a x a x a x ax a + + + = + - + - = - + = + - + - = - + + x2n - a2n = (xn - an )(xn + an ) If n is odd then, ( )( ) ( )( ) 1 2 1 1 2 2 3 1 n n n n n n n n n n n x a x a x ax a x ax a x ax ax a - - - - - - - - = - + + + + = + - + - + LL Quadratic Formula Solve ax2 + bx + c = 0 , a ¹ 0 2 4 2 x b b ac a - ± - = If b2 - 4ac > 0 -Two real unequal solns. If b2 - 4ac = 0 -Repeated real solution. If b2 - 4ac < 0 -Two complex solutions. Square Root Property If x2 = p then x = ± p Absolute Value Equations/Inequalities If b is a positive number or or p b p b p b p b b p b p b p b p b = Þ =- = < Þ - < < > Þ <- > Completing the Square Solve 2x2 - 6x -10 = 0 (1) Divide by the coefficient of the x2 x2 - 3x - 5 = 0 (2) Move the constant to the other side. x2 -3x = 5 (3) Take half the coefficient of x, square it and add it to both sides 2 2 2 3 3 5 3 5 9 29 2 2 4 4 x - x + æç - ö÷ = + æç- ö÷ = + = è ø è ø (4) Factor the left side 2 329 2 4 æç x - ö÷ = è ø (5) Use Square Root Property 3 29 29 2 4 2 x- =± = ± (6) Solve for x 3 29 2 2 x= ±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 ( ) 1 1 x , y and ( ) 2 2 x , y is 2 1 2 1 rise run m y y x x - = = - 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 ( ) 1 1 x , y is ( ) 1 1 y = y + m x - x Parabola/Quadratic Function ( )2 ( ) ( )2 y = a x - h + k f x = a x - h + k 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 = ax2 + bx + c f (x) = ax2 + bx + c The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at , 2 2 b f b a a æ æ ö ö ç- ç- ÷ ÷ è è ø ø . 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 at , 2 2 g b b a a æ æ ö ö ç ç- ÷ - ÷ è è ø ø . Circle ( x - h)2 + ( y - k )2 = r2 Graph is a circle with radius r and center (h, k ) . Ellipse ( )2 ( )2 2 2 1 x h y k a b - - + = 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 ( )2 ( )2 2 2 1 x h y k 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 that pass through center with slope ba ± . Hyperbola ( )2 ( )2 2 2 1 y k x h b a - - - = 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 slope ba ± .For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins Common Algebraic Errors Error Reason/Correct/Justification/Example 2 0 0 ¹ and 2 2 0 ¹ Division by zero is undefined! -32 ¹ 9 -32 = -9 , ( )2 -3 = 9 Watch parenthesis! ( )x2 3 ¹ x5 ( )x2 3 = x2x2x2 = x6 a a a b c b c ¹ + + 1 1 1 1 2 2 1 1 1 1 = ¹ + = + 2 3 2 3 1 x x x x ¹ - + - + A more complex version of the previous error. a bx a+ ¹ 1+ bx 1 a bx a bx bx a a a a + = + = + Beware of incorrect canceling! -a( x -1) ¹ -ax - a -a( x -1) = -ax + a Make sure you distribute the “-“! ( x + a)2 ¹ x2 + a2 ( x + a)2 = ( x + a)( x + a) = x2 + 2ax + a2 x2 + a2 ¹ x + a 5 = 25 = 32 + 42 ¹ 32 + 42 = 3+ 4 = 7 x + a ¹ x + a See previous error. ( x + a)n ¹ xn + an and n x + a ¹ n x + n a More general versions of previous three errors. ( )2 ( )2 2 x +1 ¹ 2x + 2 2( x +1)2 = 2(x2 + 2x +1) = 2x2 + 4x + 2 (2x + 2)2 = 4x2 +8x + 4 Square first then distribute! ( )2 ( )2 2x + 2 ¹ 2 x +1 See the previous example. You can not factor out a constant if there is a power on the parethesis! -x2 + a2 ¹ - x2 + a2 ( )1 -x2 + a2 = -x2 + a2 2 Now see the previous error. a ab b c c ¹ æ ö ç ÷ è ø 1 1 a a a c ac b b b b c cæ ö ç ÷ æ öæ ö = è ø = = æ ö æ ö çè ÷øçè ÷ø ç ÷ ç ÷ è ø è ø a b ac c b æ ö ç ÷ è ø¹ 1 1 a a b b a a c c b c bc æ ö æ ö ç ÷ ç ÷ æ öæ ö è ø = è ø = = æ ö çè ÷øçè ÷ø ç ÷ è ø