Algebra Review

Systems of Equations 3 * x + 5 * y = (1) x - 2 * y = (2) From (2), we can get: x = 2 y (3) Substitute (3) into (1), we can get: 3 * (2 y + 8 ) + 5 y = (4) From (4), we can get: 11 y = 8, y = 8/ (5) From (5) & (3), we can get: x = 2 * 8/ = 104/11 Standard method: solve for one variable at a time

Systems of Equations Short-cut method: work on equation as a whole x + y = (1) y + z = (2) z + x = (3) Add (1), (2), (3) together, we get: 2 (x + y + z) = 24; or x + y + z = (4) Use (4) to subtract (1), (2), (3) respectively, we get: z = 6, x = 4, and y = 2

Systems of Equations Special situation: when not enough equations available, use factor and/or property of integer. In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?

Systems of Equations In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period? Let X be population in 1991, Y in 2001, and Z in 2011: X = Y (1) X = Z (2) We got three variables X, Y and Z, but only two equations. This suggests that we need to use the property of integer.

Systems of Equations Let X be population in 1991, Y in 2001, and Z in 2011: X = Y (1) X = Z (2) From (1), we can have: Y 2 – X 2 = > (Y –X)(Y + X) = 3* 47 = 1 * (3) Since Y+X >= Y – X, we have: Y + X = 47 & Y – X = (4) or Y + X = 141 & Y – X = (5) From (4), we can get: Y = 25 & X = 22; from (2), Z = √(784) = 28 From (5), we can get: Y = 71 & X = 70; from (2), Z = 10 * √(52) invalid!

Difference of Squares a 2 - b 2 = (a + b) * (a –b) Perfect squares Polynomials (a + b) 2 = a a b + b 2 (a + b) n = a n + n * a n-1 * b + n * (n-1)/2 * a ^n-2) * b 2 + … + n * (n-1)/2 * a 2 * b n-2 + n * a * b n-1 + b n

Example: What is the 10 th digit number of ? Think of 11 = (10 + 1), we have: = (10 + 1) 2011 From (a + b) n = a n + n * a n-1 * b + n * (n-1)/2 * a ^n-2) * b 2 + … + n * (n-1)/2 * a 2 * b n-2 + n * a * b n-1 + b n we can have: (10 + 1) 2011 = * * 1 + … * 2010 / 2 * 10 2 * * 10 * Observe that only the second last term has a non-zero 10 th digit, thus the answer is 1.

Sum and product of the roots a * x 2 + b * x + c = 0 sum of the roots = - b/a product of the roots = c/a Example: The polynomial X 3 - a * X 2 + b*X – 2010 has three positive integer zeros. What is the smallest possible value of a? Let X, Y, and Z be the roots. We have: X * Y * Z = 2010 = 2 * 5 * 3 * (1) X + Y + Z = a (2) From (1), we can figure that one of the root must be some multiple of 3, 5, and 67. Hence the smallest sum of X + Y + Z = *2 = 78

Example: The roots of quadratic 3*x 2 – 7*x – 18 = 0 are m and n. Find the sum of their reciprocals. From sum & product of roots formula, we have: m + n = 7/ (1) m * n = 18/ (2) From (1) divided by (2) we can have: 1/n + 1/m = 7/ the answer

Quadratics Formula a * x 2 + b * x + c = 0 x = (-b +/- √(b 2 – 4*a*c) ) / (2 * a) (a + b) 2 = a * a * b + b 2 Perfect Square Formula

Exponents X a * X b = X (a+b) (X a ) b = X (a+b) Example: 4 a = 8, 8 b = 16, what is a*b? Note that 4 ab = (4 a ) b = (8) b = 16 We have 4 ab = 4 2 that is: a * b = 2

Example: If 1= a + a -1. What is a 2 + a -2 ? From 1= a + a -1, we can get: (a 1 + a -1 ) 2 = 1 That is: a 2 + a a * a -1 = 1 Note that: 2 a * a -1 = 2 We have: a 2 + a = 1 Thus the answer: a 2 + a -2 = -1

Use conjugate of radicals to remove square root (x + √(a) ) * (x – √(a)) = x 2 – a Example: Simplify 5 / (2 + √5) 5 5 * (2 – √5) 10 – 2 √ = = √5) (2 + √5) (2 – √5) 4 – (√5) 2 5 * (2 – √5) = = 2 √ – 5

Mean: average value of the whole set 1, 2, 3, 4, 5, 6,7, > 36 Median: the middle number, or the mean of the two middle numbers Mode: the term that occurs the most Range: difference between the highest and lowest values 1, 2, 3, 4, 5, 6,7, > 4.5 1, 2, 3, 3, 7, 7,7, > 7 1, 2, 3, 3, 5, 7,7, > 7

Arithmetic sequences a, a + k, a + 2k, a + 3k, … An = a + k * (n – 1) A1+ A2 + … + An = (A1 + An) / 2 Geometric sequences a, a*k, a*k^2, a*k^3, … Example: find the sum of 1/3 + 1/3^2 + 1/3^3 + …

Arithmetic sequences a, a + k, a + 2k, a + 3k, … A n = a + k * (n – 1) A 1 + A 2 + … + A n = (A 1 + A n ) / 2 Geometric sequences a, a*k, a*k 2, a*k 3, …

Example: find the sum of 1/3 + 1/ / / /3 5 + … Let S = 1/3 + 1/ / / /3 5 + … S = 1/3 + 1/3 (1/3 + 1/ / / /3 5 + …) S = 1/3 + 1/3 ( S ) S - 1/3 S = 1/ > 2/3 S = 1/3 S = ½

Practice Questions: 1. A rectangular parking lot has a diagonal of 10 meters and an area of 48 square meters. In meters, what is the perimeter of the parking lot? 2. Let X = … + 100, and Y = … What is the value of Y - X? 3. A majority of the 30 students in Ms. Deameanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $ What was the cost of a pencil in cents? 4. Simplify √(9 - 6* √(2) ) + √(9 + 6 * √(2) ) 5. What is the sum of all the solutions of x = |2x – | 60 – 2x | |? 6. The average of the numbers 1, 2, 3, …, 99 and X is 100 * X. What is X? 7. Positive integers a, b, and 2009, with a<b<2009, form a geometric sequence with an integer ratio. What is a?

Practice Questions: 8. For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence? 10. Let a and b be the roots of the equation x 2 – mx + 2 = 0. Suppose that a + 1/b and b + 1/a are the roots of the equation x 2 –px + q = 0. What is q? 9. Suppose that the number a satisfies the equation 4 = a + a -1. What is the value of a 4 + a -4 ? 11. Let a 1, a 2, … be a sequence for which a 1 = 2, a 2 = 3, and A n = A n-1 / A n-2 for each positive integer n>=3. What is A 2006 ? 12. Suppose that 4 a = 5, 5 b = 6, 6 c = 7, 7 d = 8, what is a*b*c*d? 13. The roots of x 2 + mx + n = 0 are twice of those of x 2 + px + m = 0. None of the m, n, and p is zero. What is the value of n/p?