Essential Question: How do we do this stuff? Chapter 2 Preview Essential Question: How do we do this stuff?

Chapter 2 Preview Use the x-intercept method to find all real solutions of the equation x3 – 8x2 + 9x + 18 = 0 Graph the function using the graphing calculator Find the roots Roots at -1, 3, & 6

Chapter 2 Preview Determine the nature of the roots 2x2 – 12x + 18 = 0 Use the discriminant to determine the number of roots: Discriminant = 0 means “1 real solution”

Chapter 2 Preview Solve by taking the square root of both sides: (4x-4)2 = 25

Chapter 2 Preview Solve by factoring: x2 + 2x – 3 = 0 Looking for two numbers that multiply to get -3 and add to get 2 Only ways to multiply to get -3 are 1 • -3 (they add to -2) -1 • 3 (they add to 2) Hey! We got a winner! Factor using those numbers (x – 1)(x + 3) = 0 Set each part of the factorization to 0 to get the solutions x – 1 = 0 or x + 3 = 0 x = 1 or x = -3

Chapter 2 Preview Solve by using the quadratic formula x2 – 2x – 5 = 0

Chapter 2 Preview Find all solutions: 5x = 2x2 - 1

Chapter 2 Preview Find all solutions: |4 – 0.2x| + 1 = 19

Chapter 2 Preview Find all solutions: |x2 - 10x + 17| = 8

Chapter 2 Preview Find all solutions:

Chapter 2 Preview Find all solutions:

Chapter 2 Preview The problem on the preview has no solution (square roots can’t ever be negative) Find all solutions:

Chapter 2 Preview Find all solutions: Real solutions? When numerator = 0 x2 + 1x – 42 = 0 (x + 7)(x – 6) = 0 x = -7 or x = 6 Extraneous solutions? When denominator = 0 x – 6 = 0 x = 6 When a solution comes up as real and extraneous, the extraneous solution takes precedence Real solution: x = -7 Extraneous solution: x = 6

Chapter 2 Preview Find all solutions: Real solutions? When numerator = 0 5x2 + 44x + 63 = 0 (5x + 9)(x + 7) = 0 x = -9/5 or x = -7 Extraneous solutions? When denominator = 0 x2 + 12x + 35 = 0 (x + 7)(x + 5) = 0 x = -7 or x = -5 When a solution comes up as real and extraneous, the extraneous solution takes precedence Real solution: x = -9/5 Extraneous solution: x = -7 or x = -5

Chapter 2 Preview Write -4 < x < 9 in interval notation If an inequality has a line underneath it, we use braces; parenthesis without. (-4, 9]

Chapter 2 Preview Solve the inequality and express your answer in interval notation: 2x – 6 < 3x + 8 [-14, ∞)

Chapter 2 Preview Solve the inequality and express your answer in interval notation: -15<-3x+3<-3 [2, 6]

Chapter 2 Preview Solve the inequality and express your answer in interval notation: Critical Points Real solutions: 5 & -9 Extraneous solution: 1 Test the intervals (-∞, -9] use x = -10, get -15/11 > 0 FAIL [-9, 1) use x = 0, get 45 > 0 PASS (1, 5] use x = 2, get -33 > 0 FAIL [5, ∞) use x = 6, get 3 > 0 PASS Interval solutions are [-9, 1) and [5, ∞)

Chapter 2 Preview The simple interest I on an investment of P dollars at an interest rate r for t years is given by I = Prt. Find the time it would take to earn $1800 in interest on an investment of $17,000 at a rate of 6.9%. You’re given I ($1800), P ($17,000) and r (6.9% = 0.069). Just plug them into the equation and solve for t 1800 / 17000 = (17000)(0.069)(t) / 17000 0.10588 / 0.069 = (0.069)(t) / 0.069 1.53 = t

Chapter 2 Preview d = -16t2 + 37. Find how long it takes the object to reach the ground (d = 0) Because time is never negative, t = 1.5 s

Chapter 2 Preview 128t – 16t2. During what period of time is the arrow above 240 feet

Chapter 2 Preview #20, continued Test the intervals (-∞, 3] -> test x = 0, get 240 < 0 FAIL [3, 5] -> test x = 4, get -16 < 0 PASS [5, ∞) -> test x = 6, get 48 < 0 FAIL The arrow is above 240 ft. from 3 to 5 sec.