Chapter(5( (Quadratic(Equations( 5.1 Factoring when the Leading Coefficient Equals 1

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1 .1 Factoring when the Leading Coefficient Equals x 6x 8 x 10x + 9 x + 10x (x )( x + 1). (x + 6)(x 4) 6. x(x 6) 7. (x + )(x + ) 8. not factorable 9. (x 6)(x ) 10. (x + 1)(x ) 11. (x + 7)(x + 4) 1. (x 6)(x ) 1. (x 11)(x + 4) 14. (x 10)(x + ) 1. (x + 4)(x + ) 16. not factorable 17. (x 9)(x + 4) 18. x(x + 1) 19. The sum will be and the other factor is The other factor is -8 and the product is -40. CK61(Algebra(II(with(Trigonometry(Concepts( 1

2 . Factoring when the Leading Coefficient Doesn t Equal 1 1. x + 9x. 6x x x 1 4. (x + )(x + ). x(x 7) 6. (x )(x +1) 7. (x + )(x 4) 8. (x + )(x + 1) 9. not factorable 10. (x 1)(8x + 1) 11. (x )(x + 1) 1. (x + 1)(x + ) 1. not factorable 14. (x )(4x + 1) 1. (x + )(10x ) 16. (x + )(x + ) 17. x(x + 7) 18. (x )(x ) 19. b = 0, (x + )(x ) 0. b = 0, a and c are square numbers, (x 4)(x + 4) CK61(Algebra(II(with(Trigonometry(Concepts(

3 . Factoring Special Quadratics 1. 1, 4, 9, 16,, 6, 49, 64, 81, 100, 11, 144, 169, 196. The sum of any two squares is not factorable, therefore, there is no formula.. (x 1)(x + 1) 4. (x + ). (4x ) 6. -(x 6) 7. (1x 7)(1x + 7) 8. (14x + ) 9. not factorable 10. (9x + ) 11. (1 x)(1 + x) 1. (11 6x) 1. not factorable 14. 4(8x 1)(8x + 1) 1. When something is squared, there are two of them. Therefore (x ) (x )(x ) =. Spencer distributed the, which is incorrect. If Spencer had FOILed (x )(x ), he would have ended up with the answer the teacher showed him. CK61(Algebra(II(with(Trigonometry(Concepts(

4 .4 Solving Quadratics by Factoring 1. x = -9, 1. x = 0, -6. x =, 4 4. x =, 4. x =, - 6. x = ; double root 7. not factorable 8. x = 0, 1 9. x = 10. x = 4, -8 7, not factorable 1. x = -, 1. x = 14. x =, 4 1, 7 1. x = 1 6, 16. The length is 64 feet, the width is 9 feet. CK61(Algebra(II(with(Trigonometry(Concepts( 4

5 . Simplifying Square Roots CK61(Algebra(II(with(Trigonometry(Concepts(

6 .6 Dividing Square Roots CK61(Algebra(II(with(Trigonometry(Concepts( 6

7 CK61(Algebra(II(with(Trigonometry(Concepts( 7

8 .7 Solving Quadratics using Square Roots 1. x = 1, -1. x =, -. x = ± 4. x = -9,. x = ± 6. x = ± 1 7. x = 4, x = ± x = 7± x = -, x = 10 ± 6 1. x = 1± 7 1. x = 11, 14. x = ± 1. x = 4,$ 16. will vary. 17. x =, -, opinions will vary. 18. x =, ; the answers are the same for both quadratics. This means the quadratic equations are the same (or reduce to be the same). 19. w = 6.61, h = seconds CK61(Algebra(II(with(Trigonometry(Concepts( 8

9 .8 Defining Complex Numbers 1. i. 11i. 18i 4. 4i i i 10. i 11. -i 1. i i i i i i 18. 1i i i CK61(Algebra(II(with(Trigonometry(Concepts( 9

10 .9 Multiplying and Dividing Complex Numbers 1. i i. -8 i i i i i i i i i 4 i 1 6 i i CK61(Algebra(II(with(Trigonometry(Concepts( 10

11 .10 Solving Quadratic Equations with Complex Number Solutions 1. x =± i. x =± i. x = 1± 11i 4. x =± i. x =± i 6 6. x = 9± 4i 7. x = 6± i 8. x = ± i x = ± i x = ± i x = ± i i 1. 6 i 14. x 6x x x + CK61(Algebra(II(with(Trigonometry(Concepts( 11

12 .11 Completing the Square when the Leading Coefficient Equals ( x + ) 7 x 1 x 4 7. x = ± 6 8. x = ± i 9. x = 7± i x = ± i 9 19 x = ± 1. x = 10± x = -6, 14. x = 9± i 1. x = -7, x = ± CK61(Algebra(II(with(Trigonometry(Concepts( 1

13 17. x = ± i 18. x = -1, x = 11, -4; opinions will vary x = ± i CK61(Algebra(II(with(Trigonometry(Concepts( 1

14 .1 Completing the Square when the Leading Coefficient Doesn t Equal 1 1. x = 1± x = ± 4. x = ± i 4. x =,$. x = 7± i x =,$ 7. x =, x =,# 1 9. x = 4± i x = ± 1 x =,$ 0 x = ± i ± x = 1 x =,$ 4 b b 4ac a 1. yes, (4x )(x + 1) 16. In the third step, they should have added 144 to the right side, 4 %6, no just 6. The correct answer is 1 x = 6 ±. CK61(Algebra(II(with(Trigonometry(Concepts( 14

15 .1 Deriving and Using the Quadratic Formula 1. x = 4± x = 4,$ 4 1± 41 x = x = ± i x = ± i 6 6. x = 7; double root 7. x = 10, x =,$ x = ± i 1 x =,$ 11. x = 8± x =± 1. x = ; double root 14. x = 1, - 1. will vary. In general, students should try to factor an equation first. If it cannot be factored, then they should either complete the square or use the Quadratic Formula. Both of the later options work if the solutions are imaginary or irrational. CK61(Algebra(II(with(Trigonometry(Concepts( 1

16 .14 Using the Discriminant 1. 1 real solution. real solutions. imaginary solutions 4. real solutions. real solutions 6. imaginary solutions 7. x = 0, - 8. x = x = ± i c< 1, c = 1, c > c< 9, c = 9, c > 9 1. c< 6, c = 6, c > k k> and k < - 1. k = and -, < k < CK61(Algebra(II(with(Trigonometry(Concepts( 16

17 .1 Finding the Parts of a Parabola 1. (6, -); minimum. (-, -4), minimum. 4. 1,$18 ; maximum 1 1,$ 1 ; minimum 4 8. (, 0); maximum 6. (16, 1); maximum 7. (6, -), (11, 0), (1, 0), (0, 11), x = ,#4, (-4, 0,), 4,#0, (0, 1), x = 4 9. (-6, -7), (-1, 0), (, 0), (0, -1), x = ,$ 6, (-9, 0), 1,#0, (0, -9), x = (, 0), (, 0), (0, -), x = 1. 1,$40, (14, 0), (-4, 0), (0, 8), x = 1. Quadratic Formula or completing the square. 14. (4, 7), ( 4+ 7,$0), ( 4 7,$0) ,# 8, ,%0, 4 1 6,% ,, 1,, , 1, 9, 1, 4 CK61(Algebra(II(with(Trigonometry(Concepts( 17

18 18. These two parabolas do not have any x-intercepts, which means they do not have real solutions. The solutions for these quadratic equations are imaginary. For #16, x = 1 ± i. 19. A parabola can intersect the x-axis three different ways;, 1, or 0. If it intersects it twice, there are two real solutions. If it intersects the x-axis once, then the vertex is the only solution (repeated root; #11). If it does not intersect the x-axis at all, then there are no real solutions or two imaginary solutions (#16 and #17). ans y = b + 4ac 4a CK61(Algebra(II(with(Trigonometry(Concepts( 18

19 .16 Vertex, Intercept, and Standard Form 1. Intercepts (or how to find the Equation Vertex intercepts) Standard y = ax + bx + c b b Factor or use the Quadratic Form, f Formula to find the intercepts. a a Intercept y = a( x p)( x q) Form p + q, f p + q (p, 0) and (q, 0) Vertex Form y = a( x h) + k (h, k) Continue to solve equation using square roots.. (4, -9), (7, 0), (1, 0). (1, -49), (-6, 0), (8, 0) 4. (-1, -9), (, 0), (-4, 0). (-1, 6), (, 0), (-7, 0) 6. (-1, -), + 6,$0, 6,$0 7. (, 4), other points: (0, 16), (4, 16) 8. (, -1), (9, 0), (-, 0) 9. (-, 7), ( + 7,$0), ( 7,$0) ,%, (4, 0), 8 16,# y = (x )(x 1) 1. y = -(x + 1)(x ) 1. 1 y = 4 x + ( x + 4) CK61(Algebra(II(with(Trigonometry(Concepts( 19

20 y = + ( x 6) 8 y = + ( x ) 1 y = + ( x ) 7 y x x = y = x x = 6 9 y x x CK61(Algebra(II(with(Trigonometry(Concepts( 0

21 .17 Using the Graphing Calculator to Graph Quadratic Equations 1. (0., -6.), (-, 0), (, 0). (1., 0.), (-4, 0), (7, 0). (-.7, -.1), (-8, 0), (., 0) 4. (, -), (1.9, 0), (4.41, 0). (-4, -), (-.7, 0), (-.7, 0) 6. (-, ), x = ± i 7. (0.6, -7.), (-0., 0), (1., 0) 8. (1., 6), (0.8, 0), (.7, 0) 9. (4, 1), x = 4± 6i 10. (-1, -9), i x = 1± 11. a changes the width (or breadth) of the parabola. If a > 1, then the parabola is wider than y = x. If 0 < a < 1, then the parabola will be narrower. 1. If a is negative, then if flips the parabola upside-down. 1. h shifts the parabola to the right or left. If h is negative in the equation, it will shift the parabola to the right. If it is positive, it will shift the parabola to the left. 14. k shifts the parabola up and down. If k is negative in the equation, it will shift the parabola down. If it is positive, it will shift the parabola up. 1. The maximum height is the y-coordinate of the vertex or feet. The ball travelled a total distance of 46.6 feet. CK61(Algebra(II(with(Trigonometry(Concepts( 1

22 .18 Modeling with Quadratic Functions y = + + ( x 1) 1 1 y = ( x + )( x ) 4 y = ( x 9) 4 1 y = ( x 8)( x + ) y = ( x + 9)( x + 7) y = + ( x 6) 10 4 ( 4) x 1 y = + 9 y 7 8 = x + y = ( x )( x 16) y = x + x 1 y = x + x = y x x 1. See Complete Solution Key y = 0.07x + 1.x y = x + x a) y x x = b) willvary. CK61(Algebra(II(with(Trigonometry(Concepts(