GMAT (Subject) / Quantitative - Algebra (Lesson)

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• (x+y+z)^2 =x2+y2+z2+2(xy+yz+zx)
• PEMS Pattern Parantheses ll () Exponents Multiply/Division :/ Substrat/Addition +-
• SOLVE xyz Three equations 1. Substitute One Expression 2. Combine the Last Expression via Solving System x  y  z  = 8 1  0  1   = 5 0  1  1    = 3
• Factor Out a Common Term 113+114 => 113 * (11+1) = 113 * (12)
• Fractional Exponents Numerator: Powert to Raise to Denominator: Root to Take
• Compound Base 103 = (2*5)3 = 23 * 53 =8*125
• SOLVE Exponential Equations Rewrite either the exponent or the base, so that the same exponent or the same base appears => Same Base  Equal only Exponents
• Useful Squares to know 11^2 = 121 12^2 = 144 13^2 = 169 14^2 = 196 15^2 = 225 16^2 = 256  20^2 = 400
• Two Solution Equations lw-yl = 8 1. Isolate term lw-yl = 18 2. Remove Brackets  w-y = 18 CASE 1: w-y = 18 CASE 2: -(w-y) = 18 3. CHECK IT
• Operation Expressions: PULL OUT A COMMON TERM 2ab+4b = 2b (a+2) 
• Negative Exponents One under the same thing with a positive exponent y2 / y5 => y-3 = 1/y3 x/y -2 = y/x 2 
• SIGNS in Equations with EVEN OR ODD Exponents EVEN Exponents HIDE the +- Sign of the Base => all nrs become positive ANY EVEN EXPONEND in an equation make it dangerous x2 = 25 <-> lxl = 5 -> x = +5; -5 FOR ANY x, √x2 = lxl ODD EXPONENTS KEEP the sign of the Base
• If m and n are positive integers and 2^18 * 5^m = 20^n, what is the value of m?2 218*5m = 20n 218 * 5m = 22n * 5n Because m and n are integers, there must be the same number of 2s and the same number of 5s on either site 18= 2n  n=9 m=n
• ROOTS in Equations with ODD and EVEN ROOTS EVEN ROOTS HIDE the sign of => positive or negative x ODD ROOTS KEEP the sign of x
• Simplify a ROOT √25*16 = √25 * √16 = 5*4 = 2ß ONLY POSSIBLE IN MULTIPLY/DIVISON
• Imperfect Squaren √52 = √2*2*13 = 2*√13 √72 = √2*2*18 = √2*2*2*3*3* = 2*√2*3 = 6*√2
• QUADRATIC EQUATIONS: FACTORING (ax2+Bx-C) = 1. PRODUCT C => see what it is made of 2. SUM B => Test if it equals the Sum
• HIGHER ORDER EQUATION x3 + 2x2 -3x = 0 1.) Factor out 1 =>  x (x2+2x-3) = 0  1b.) X=0 is one Solution!  2.) FACTORING x* ((x-1) (x+3)) = 0 2b) 1 and -3 are more solutions
• a2+b2=116 and we know a,b are integers such that a<b We can solve this by testing values of a and checking if we can find b a=1 b=root(115) Not integer a=2 b=root(112) Not integer a=3 b=root(107) Not integer a=4 b=root(100)=10 a=5 b=root(91) Not integer a=6 b=root(80) Not integer a=7 b=root(67) Not integer a=8 b=root(52)<a So the answer is (4,10)
• Reducing to lower degree by Substituion This is useful sometimes when it is easy to see that a simple variable substitution can reduce the degree => x6−3x3+2 = 0 Here let y=x3 y2−3y + 2=0 (y−2)(y−1)=0 So the solution is y=1 or 2 RESUBSTITUTE x^3=1 or 2 or x=1 or 2‾√3
• Integers Other tricks Sometimes we are given conditions such as the variables being integers which make the solutions much easier to find. When we know that the solutions are integral, often times solutions are easy to find using just brute force.
• Factorization This is the easiest approach to solving higher degree equations. The basic idea is that if you can write an equation in the form : A∗B∗C=0 Eg. x3+11x2+30x=0 x∗(x2+11x+30)=0 x∗(x+5)∗(x+6)=0 So the solution is x=0,-5,-6
• The general form of a quadratic equation is ax2+bx+c=0. When will there be no solution? The equation has no solution if b2 < 4ac The equation has exactly one solution if b2=4ac This equation has 2 solutions if b2 > 4ac
• How to get an absolute value back ? In Algebraic Identities : These are generalized relationships such as √x2=|x|f you square it and take the root, you get the absolute value back. So the variable acts like a true placeholder, which may be replaced by any number.
• If each number in a sequence is three numbers more than the previous number and the sixth number is 32, what is the 100th number? 1. FIND NUMBER OF JUMPS = 100-6=94 2. INCREASE per JUMP= 3  3. TOTAL INCREASE = 94*3 + 32
• SEQUENCES AND PATTERNS What is the unts digit of S65 When Sn = 3n 31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 PATTERNS: 3971 3971 => as 65 end with a 5 => the digit is a 3 YOU MAY ALSO LOOK AT EVEN AND UNEVEN NUMBERS
• Multiple absolute values Multiple absolute values should be dealt with by working with the innermost one and moving outwards. For example: | 3 - (| -5|) |= |3 - 5|= |-2|= 2
• If lx-2l = l2x-3l, what are the possible values for x? The positive/positive case:    (x-2) = (2x-3) The positive/negative case:   (x-2) = -(2x-3) negative/positive case:         -(x-2) = (2x-3) negative/negative case:        -(x-2) = -(2x-3)
• IF x= 9b-3ab / (3/a - a/3) - what is x? x= 9b-3ab / (3/a - a/3) 1. Same Denominator: x= 9b-3ab /(9-a2/3a) 2. Flip Bottom Fraction: x= 3a*(9b-3ab) / (9-a2) =>  3. Factore out: x= (3a)*(3b)*(3-a) / (3-a)*(3+a) 4. Cancel out: x= (3a)*(3b)/ (3+a) = 9ab/3+a
• x^ a+b / x^b = x^ a+b / x^b = x^ a+b-b = xa
• A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 mil to a population of 40 mil in one hour, by what factor does the population increase every 10 minutes? Exponential growth problem:  y(t) = y0 * kt 1. growth factor in one hour : 40mil / 5 mio = 8  2. k6 = 8 3. k = 81/6
• For which of the following functions does f(x) = f (2-x) Symmetrie function type of problem 1.) pick an example, e.g. x=4                                 -----> f(4)                            f ( 2-4) = f(-2) 2.) f(x) = x+2            -----> f(4)=6                      f (-2)= -2+2 = 0 => klappt nicht 3.) weiter ausprobieren!!
• Which of the following equations has no solution for a? a^2 - 6a + 7 = 0 ... a^2 - 6a - 7 = 0 ... 1.) look for the negative discriminant: 2.) DANACH SUCHEN         - b2- 4ac <0 ??
• If 4/x < 1/x, what is the possible range of values for x? Case x>0 4/x < 1/3 12<x Case x<0 4/x < 1/3 12>x Hence, x<0 OR x>12
• (1+√5)(1-√5) = = (1+√5)(1-√5) = 1^2 - √5 ^2 =  (..+..)(..-..) wird immer zu 12-x2
• When 2^r*4^s=16, then 2r+s = 1. Exponent.rule  2r*4s = 16 2r * 22s = 24 ALSO r+2s = 4 2. Interpretieren. -> STICHWORT POSITIVE INTEgEr Wenn r+2s = 4; dann muss s>2, also s=1 ... und r = 2 3. 2r+s = 2*2 +1 = 5
• Three people each contribute x dollars to the purchase of a car. They then bought the car for y dollars, an amount less than the total number of dollars contributed. If the excess amount is to be refunded to the three people in equal amounts, each person should receive a refund of how many dollars? Total refunded = Total contributed - price of the car Refund p. person = Total refund / 3
• A certain fishing boat is chartered by 6 people who are to contribute equally to the total charter cost of 480 $. if each person contributed equally to a $ 150 down payment, how much of the charter cost will each person still owe? Since each of the 6 individuals contribute equally to the 150$ down payment, and since it is fiven that the total cost of the chartered boat is 480$, each person still owes ($480- $150) / 6 = $55 Down payment = Anzahlung
• How is it possible to get a units digit of 9? If x*2's unit digit is 9, and the unit digit of (x+1)*2 is 4, what is the unit digit of (x+2)*2? 1. A unit digit of 9 is only possible, is x is 3 or 7 2. PLUS in 3 or 7 and see if the second equation fits with the assumption
• "Does k have a factor p?" TRANSLATION: Is k a prime number or not?
• For which of the following does f(x) = f (1/x), given that x is not -2;-1;0;1 ? 1. PICK A POSSIBLE NUMBER X=3 2. MAKE A CHART                                                       f(3)                     f(1/3) 3. ANWER CHOICES A) l x+1 l / lxl B)  C) 4. PLUG IN ANWER CHOICES AND TRY OUT
• For which of the following functions does f(x-y) NOT EQUAL f(x) - f(y), given that f(y) is f(2-x) 1. PICK A POSSIBLE NUMBER x=4 2. MAKE A CHART                                                      f(4)                             f(2-4) = f(-2)  ANWER CHOICES A) X+2                                      ____TRY OUT____ B) 2X-X2 C) 2-X
• EXTREME VALUES : 2y+3 =< 11 and 1<x<5, what is the maximum possible value for xy? 1. REPHRASE EQUATIONS Y=< 4 and 1<X<5 2. SET UP CHART                        Extreme Value for X            Extreme Value for Y               Extreme Value for XY max                            5                                           4                                             20 min                              1                                     endless                                     endless                             
• Which of the following equations is NOT equivalent to 10y^2=(x+2)(x-2) NOTE: The equation becomes 0 when x= 2;-2 Every equation that is equivalent holds the same property ! CHECK: for which equation does it not become 0
• If a(a+2) = 24 and b(b+2)=24 and a is not b, then a+b = SECOND DEGREE EQUATIONS - FACTORING a(a+2) = 24 given a2 + 2a = 24 a2 + 2a - 24 = 0 (a+6) * (a-4) = 0 => a= -6 or a = 4 As for b, there is the same solution, so either a or b equals 4 or -6.  a+b is hence 4-6 = -2
• If the smallest positive integer y such that 3150 multiplied by y is the square of an integer, then y must be ? to find the smalles positive integer y such that 3150 y is the square of an integer, first find the prime fcatorization of 3150 by a methos similar to the following:  3150 = 10*315 = 2*5 * 3*105 = 3*5*3*5*3*7 = 32*52*2*7 To be a PERFECT SQUARE, 3150y must have n even number of each of its prime factors. At minimum, y must have on factor of 2 and one factor of 7, so that has 2 factors of each of its prime factors.  142 * 32 * 52 => the answer is 2*7 = 14
• Which of the following equations has 1+√2 as one of its roots? The problem can be solved by working backwards to construcs a quadratic equation with 1+√2 as a root that does not involve radicals.  x = 1+√2        l set x to the desired value x-1 = √2        l subtract 1 from both sides (x-1)2 = 2      l square both sides x2-2x+1 = 2  l expand the left side x2-2x-1 = 0   l subract 2 from both sides
• Non-negative numbers Positive: >0 Non-negative: ≥0
• Powers of even and odds If the base is positive, raising it to an odd power will give a positive result as well. e.g. 23 = +8 if the base is negative, raising it to an odd power will give a negative result. e.g. (-2)3 = (-2)·(-2)·(-2) = -8
• Powers even and odd An even power is always positive, whether the base is positive or negative. An odd power retains the base's original sign. 

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