Algebra 2 Index Page 2 9 Page 10

Algebra 2 Index Page 2 – 9 Page 10 – 12 Page 13 Page 14 Page 15 - 17 Page 18 - 19 Page 20 – 24 Page 25 – 33 Page 34 + 17: 29 Quadratics (guide number method) Algebra fractions (with quadratics) -b formula Other types of factorising Simultaneous Equations (non linear) Finding equations from routes Rules of Indices Rules of Surds Exam Question Mr. D. Mc. Carthy

List the paired factors of the following numbers. . . 16 1 x 16 2 x 8 4 x 4 Also. . . -1 x -16 -2 x -8 -4 x -4 17: 29 36 1 x 36 2 x 18 3 x 12 4 x 9 6 x 6 Also. . . -1 x -36 -2 x -18 -3 x -12 -4 x -9 -6 x -6 -12 1 x -12 2 x -6 3 x -4 Also -1 x 12 -2 x 6 -3 x 4 Mr. D. Mc. Carthy -20 25 1 x -20 2 x -10 4 x -5 Also -1 x 20 -2 x 10 -4 x 5 1 x 25 5 x 5 Also -1 x -25 -5 x -5

Factorise a 2 – 2 ab + ac – 2 bc x 2 – 3 x + xy – 3 y a 2 – 2 ab + ac – 2 bc a(a – 2 b) + c(a – 2 b) (a + c)(a – 2 b) x 2 – 3 x + xy – 3 y x(x – 3) + y(x – 3) (x + y)(x – 3) ab – bc + ac - b 2 ab + ac - bc – b² a(b + c) - b(c + b) (a - b)(c + b) 17: 29 3 ax – 6 ay – 4 by + 2 bx 3 a(x – 2 y) + 2 b(-2 y + x) (3 a + 2 b)(x – 2 y) Mr. D. Mc. Carthy

Guide Number Method x² + 13 x + 22 GN = 22 1 x 22, -1 x -22 2 x 11, -2 x -11 x² + 2 x + 11 x + 22 x(x + 2) + 11(x + 2) (x + 11)(x + 2) 17: 29 x² - 8 x - 48 GN = -48 1 x -48, -1 x -22 2 x -24, -2 x -11 4 x -12, -4 x 12 6 x -8, -6 x 8 x² + 4 x - 12 x - 48 x(x + 4) - 12(x + 4) (x - 12)(x + 4) Mr. D. Mc. Carthy

More difficult ones 4 x² - 29 x + 7 G. N. = 4 x 7 GN = 28 1 x 28 -1 x -28 2 x 14 , -2 x -14 4 x² - 1 x - 28 x + 7 x(4 x – 1)- 7(4 x – 1) (x – 7)(4 x – 1) 17: 29 6 x² - 17 x - 3 G. N. = 6 x -3 GN = -18 1 x -18 -1 x 18 2 x -9 , -2 x 9 3 x -6 -3 x 6 6 x² + 1 x - 18 x - 3 x(6 x +1)- 3(6 x + 1) (x – 2)(6 x + 1) Mr. D. Mc. Carthy

Exercise Q 10 and Q 11 10. 6 x² - x - 2 = 0 GN = -12 Factors -4 x 3 6 x² - 4 x + 3 x - 2 = 0 2 x(3 x – 2) + 1(3 x – 2) (2 x + 1)(3 x – 2) = 0 2 x + 1 = 0 3 x – 2 = 0 x = -½ x=⅔ 17: 29 11. 4 x² - 29 x + 7 = 0 GN = 28 Factors -28 x -1 4 x² - 28 x – x + 7 = 0 4 x(x – 7) – 1(x – 7) (4 x – 1)(x – 7) = 0 4 x – 1 = 0 x – 7 = 0 x=¼ x=7 Mr. D. Mc. Carthy

Homework 9. 2 x² - 5 x +2 = 0 GN = 4 Factors -4 x -1 2 x² - 4 x – x +2 = 0 2 x(x – 2) – 1(x – 2) (2 x – 1)(x – 2) = 0 2 x – 1 = 0 x – 2 =0 x=½ x=2 17: 29 15. 6 x² + 17 x - 3 = 0 GN = -18 Factors 18 x -1 6 x² + 18 x – x - 3 = 0 6 x(x + 3) – 1(x + 3) (6 x - 1)(x + 3) = 0 6 x - 1 = 0 x + 3 = 0 x = ⅟₆ x = -3 Mr. D. Mc. Carthy

Q 35 2 xcm A B xcm 2 xcm Area of A (2 x) = 4 x² Area of B (4). (x) 17: 29 = 4 x 4 cm Total area = 4 x² + 4 x We know the area = 48 cm² 4 x² + 4 x = 48 4 x² + 4 x – 48 = 0 ÷ 4 x² + x – 12 = 0 (x + 4)(x – 3) = 0 x = -4 x = 3 x=3 Mr. D. Mc. Carthy

(x + 6)cm xcm Q 32 (x + 3)cm 16 cm Area of Rectangle Area = length x breadth A = x(x + 6) AR = x² + 6 x We know AR = AT x² + 6 x = 8 x + 48 x² 17: 29 - 2 x – 48 = 0 Area of Triangle Area = ½(base x height) A = ½[16(x + 3)] A = ½(16 x + 48) AT = 8 x + 48 x² - 2 x – 48 = 0 (x – 8) (x + 6) = 0 x=8 x = -6 Mr. D. Mc. Carthy x=8

Algebraic Fractions Always get rid of the fraction first 5 +3–x = 5 3 x 4 x Find LCM of numbers under the line (3 x). (4 x) 5 + (3 x). (4 x) (3 – x) = 5(3 x). (4 x) 3 x 4 x 4 x(5) + 3 x(3 –x) = 5(3 x). (4 x) 20 x + 9 x - 3 x² = 60 x² + 3 x² -20 x – 9 x = 0 63 x² - 29 x = 0 17: 29 Mr. D. Mc. Carthy Find LCM of: (i) 6 and 4 = 12 (ii) 5 and 9 = 45 (iii) 13 and 21 To find a COMMON multiple we multiply one by the other 13 x 21 = 343

More complicated. . (as complicated as it gets) 3 2 = 4 (x – 3) (2 x + 1) 3 LCM of numbers under the line: (x – 3)(2 x + 1)(3)3 - (x – 3)(2 x + 1)(3)2 = 4(x – 3)(2 x + 1)(3) (x – 3) (2 x + 1) 3 (2 x + 1)(3)(3) - (x – 3)(3)(2) = 4(x – 3)(2 x + 1) [18 x +9] - [6 x - 18] = [8 x² - 20 x – 12] 18 x + 9 - 6 x + 18 = 8 x² - 20 x – 12 8 x² - 32 x – 39 = 0 Solve quadratic after that. . . 17: 29 Mr. D. Mc. Carthy

4 + 1 =3 y y– 1 LCM = (y)(y – 1) Exercise (y)(y – 1)4 + (y)(y – 1)1=3(y)(y – 1 y y -1 4(y – 1) + y = 3 y(y – 1) 4 y – 4 + y = 3 y² - 3 y – 4 y - y + 4 = 0 3 y² - 7 y + 4 = 0 GN = 12 Factors = -6, -2 3 y² - 6 y - 2 y + 4 = 0 3 y(y – 2) – 2(y – 2) (3 y – 2)(y -2) = 0 y = 17: 29 ⅔ y=2 LCM = (x)(x+2)(2) (x)(x+1)(2)(x – 1) + (x)(x+1)(2)(x) x x+1 2(x + 1)(x – 1) + 2(x)(x) = (2 x + 2)(x – 1) + 2 x² = x² + x = (x)(x+1)(2)1 2 x(x + 1) 2 x² - 2 x + 2 x – 2 + 2 x. ² = x² + x 3 x² - x – 2 = 0 GN = -6 Factors = -3 and 2 3 x² - 3 x + 2 x – 2 = 0 3 x(x – 1) + 2(x – 1) = 0 Mc. Carthy (3 x. Mr. + D. 2)(x – 1) = 0 3 x + 2 = 0 x = -⅔ x– 1=0 x=1

4 x² - 5 x -2 = 0 a = 4 b = -5 Split up into: c = -2 and x = 1. 57 17: 29 Mr. D. Mc. Carthy or x = - 0. 32

Exercise (i) 4 x² - 12 x + 5 Solve the following (ii) 5 x² - 25 x = 0 GN = 20 Take out what’s common Factors = -10 and -2 5 x(x – 5) = 0 4 x² - 10 x – 2 x + 5 = 0 5 x = 0 x– 5=0 2 x(2 x – 5) – 1(2 x – 5) x=0 x=5 (2 x – 1)(2 x – 5) = 0 x=½ x = 5/2 17: 29 Mr. D. Mc. Carthy (iii) 25 x² - 16 = 0 Difference of 2 squares (5 x + 4)(5 x – 4) = 0 5 x + 4 = 0 5 x – 4 = 0 x = -4/5 x = 4/5

Simultaneous Equations A way to solve for x and y using two equations Example: 3 x – 2 y = 7 It is impossible to know for certain what x and y are. If we have two equations however. . . 3 x – 2 y = 7 4 x + 4 y = -13 We can solve using simultaneous equations 17: 29 Mr. D. Mc. Carthy

Solving Quadratic Simult. eqs Solve for x when x - 2 y = 3 and x² - y² =24 x – 2 y = 3 3 y² + 12 y - 15 = 0 GN = -45 x = 2 y + 3(Fill this into other equation) Factors = 15, -3 3 y² + 15 y – 3 y - 15 = 0 x² - y² =24 3 y(y + 5) – 3(y + 5) (3 y – 3)(y + 5) = 0 (2 y + 3)² - y² = 24 3 y – 3 = 0 y + 5 = 0 4 y² + 12 y + 9 - y² = 24 y = 1 y = -5 x – 2 y = 3 3 y² + 12 y - 15 = 0 x – 2(1) = 3 x – 2(-5) = 3 x– 2=3 x +10 = 3 Solve quadratic x=5 x = -7 17: 29 Mr. D. Mc. Carthy

Exercise Solve for x and y in the following (i) x + y = 1 (ii) x + 3 = 2 y x² + y² = 13 xy – 7 y + 8 = 0 x+y=1 x=1–y x² + y² = 13 (1 – y)² + y² = 13 1 + y²- 2 y + y² = 13 2 y² - 2 y – 12 = 0 y² - y – 6 = 0 (y – 3)(y + 2) = 0 y = 3 y = -2 x+3=1 x = -2 17: 29 x + (-2) = 1 x– 2=1 x=3 x + 3 = 2 y x = 2 y – 3 xy - 7 y + 8 = 0 (2 y - 3)y - 7 y + 8 = 0 2 y² - 3 y – 7 y + 8 = 0 2 y² - 10 y + 8 = 0 y² - 5 y + 4 = 0 (y – 4)(y – 1) = 0 y=4 y=1 Mr. D. Mc. Carthy x + 3 = 2(4) x=8– 3 x=5 x + 3 = 2(1) x=2– 3 x = -1

Working Backwards Solve the following Quadratic 4 x² - 12 x + 5 = 0 GN = 20 Factors: -10, -2 4 x² -10 x - 2 x + 5 = 0 2 x(2 x – 5) -1(2 x – 5) = 0 (2 x -1)(2 x – 5) = 0 2 x - 1 = 0 2 x = 1 x=½ 17: 29 2 x - 5 = 0 2 x = 5/2 Mr. D. Mc. Carthy x = ½ and x = 5/2 are called the roots of this equation. If we are given the roots at the start of the equation. . . Eg. the roots of an equation are 4 and -3. Find the equation. x = 4 x = -3 x– 4=0 x+3=0 (x – 4)(x + 3) = 0 x(x + 3) – 4(x + 3) = 0 x² + 3 x – 4 x – 12 = 0 x² - x – 12 = 0

Exercise Work out what equations can be made with the following roots (i) 5 and -4 (ii) ⅔ and -¾ x = 5 x = -4 x=⅔ x = -¾ x– 5=0 x+4=0 3 x = 2 4 x = -3 3 x – 2 = 0 4 x + 3 = 0 (x – 5)(x – 4) = 0 x(x + 4) – 5(x + 4) (3 x – 2)(4 x + 3) x² +4 x - 5 x -20 3 x(4 x + 3) – 2(4 x + 3) x² - x -20 12 x² + 9 x - 8 x – 6 If the roots of the equation 12 x² + x – 6 x² + ax +b are 5 and -4, then work out what a and b are. a 17: 29 = -9 b = -4 Mr. D. Mc. Carthy

Indices Work out the following 4⁴ x 4³ 3³ x 3² = 16384 = 243 = 3⁵ = 4⁷ 6⁵ ÷ 6³ = 36 = 6² 17: 29 7⁴ ÷ 7³ =7 = 71 Mr. D. Mc. Carthy

Laws of Indices 1. = eg. 5⁴ x 5⁸ = 512 an x am an + m 2. an = an - m am eg. 5⁹ = 5⁴ 5⁵ 5. a⁰ = 1 4. abm = am x bm eg. 12⁴ = 6⁴ x 2⁴ Also applies with routes. . . 17: 29 3. (am)n = amn eg. (6²)⁴ = 6⁸ Anything to the power of 0 = 1 1343523⁰ = 1 Mr. D. Mc. Carthy Q 1 -2 page 38

More rules 2. an = an - m am eg. 5⁹ = 5⁴ 5⁵ 4½ = √ 4 = 2 27 ¹ ₃ = ∛ 27 = 3 17: 29 6⁵ 6⁸ = 6 -3 = 1 63 3. (am)n = amn eg. (6²)⁴ = 6⁸ Mr. D. Mc. Carthy 64⅔ = (64¹ ₃)2 (∛ 64)2 42 = 16

Exercise Work out what x is 4 x = 64 4 x = 43 x=3 3 x = 81 81 = 9²= (3²)2 = 34 x=4 49 x = 7 49½ = 7 x=½ 2 x = ¼ 17: 29 Mr. D. Mc. Carthy

Homework Q 6 (i) 6 m⁹ = 2 m⁷ 3 m² 2 6 xmxmxmxmxm 3 xmxm = 2 m⁷ 17: 29 Mr. D. Mc. Carthy

LCM = x x(x) – x(1) = 2(x) x x² - 1 = 2 x x² -2 x - 1 = 0 -b formula 17: 29 a=1 Mr. D. Mc. Carthy b = -2 c = -1

Surds 17: 29 Rules (multiplication/division): √ab =√a. √b Applications: √ 45 = √ 9. √ 5 = 3√ 5 (in its simplest form) But also. . . √ 8. √ 4= √ 32 Example: 5√ 3 x 4√ 6 20√ 18 = 20. √ 9. √ 2 = 20. 3. √ 2 = 60√ 2 Mr. D. Mc. Carthy

Rules Ctd. Hint: Treat similar to algebra 5√ 6 + 4√ 6 = 9√ 6 If the number is the same you can add as normal Example √ 3 + √ 12 + √ 27 √ 3 + √ 4. √ 3 + √ 9. √ 3 + 2√ 3 + 3√ 3 = 6√ 3 17: 29 Mr. D. Mc. Carthy

Exercise Find in its simplest form: √ 16. √ 2 4√ 2 √ 4. √ 5 2 √ 5 Simplify (3 x – 6)(4 x + 12) 3 x(4 x + 12) – 6(4 x + 12) 12 x² + 36 x – 24 x – 72 12 x² + 12 x – 72 17: 29 √ 9. √ 8 √ 100. √ 2 3 √ 8 10 √ 2 3 √ 4. √ 2 3. 2√ 2 6 √ 2 (3 a – 3 b)(4 a + 5 b) 3 a(4 a + 5 b) – 3 b(4 a + 5 b) 12 a² + 15 ab – 12 ab – 15 b² 12 a² + 3 ab – 15 b² Mr. D. Mc. Carthy

Exercise Q 4, 8, 10 page 45 4+ 17: 29 3 x= Mr. D. Mc. Carthy

Exercise Q 16 and Q 17 (x + √x)(x - √x) x(x - √x) + √x(x - √x) x² - x√x + x√x – x x² - x Hence. . . 17: 29 (x + √x)(x - √x) = 6 x² - x – 6 = 0 (x – 3)(x + 2) = 0 x– 3=0 x+2=0 x=3 x = -2 Mr. D. Mc. Carthy

Q 17 (√x – 2) (√x + 2) √x √x 17: 29 Mr. D. Mc. Carthy

Solve and leave in the form a ± √b x² – 6 x + 3 = 0 17: 29 Mr. D. Mc. Carthy

Exam Questions 17: 29 Mr. D. Mc. Carthy

17: 29 Mr. D. Mc. Carthy

b(i) (ii) x+y=7 x² + y² = 29 x+y=7 x=7–y (7 – y)² + y² = 29 49 – 14 y + y² = 29 2 y² - 14 y + 20 = 0 y² - 7 y + 10 = 0 (y – 5)(y – 2) = 0 y=5 y=2 17: 29 x+5=7 x=2 x+2=7 x=5 6 – 2 y < 0 6 – 2(5) 6 – 2(2) = 6 – 10 6– 4 = -4 =2 _____ y=5 Mr. D. Mc. Carthy

2008 17: 29 Mr. D. Mc. Carthy

2006 17: 29 Mr. D. Mc. Carthy