Chapter 2 Preview

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

2. Chapter 2 Preview • Use the x-intercept method to find all real solutions of the equationx3 – 8x2 + 9x + 18 = 0 • Graph the function using the graphing calculator • Find the roots • Roots at -1, 3, & 6

3. Chapter 2 Preview • Determine the nature of the roots2x2 – 12x + 18 = 0 • Use the discriminant to determine the number of roots: • Discriminant = 0 means “1 real solution”

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

5. 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

6. Chapter 2 Preview • Solve by using the quadratic formulax2 – 2x – 5 = 0

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

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

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

10. Chapter 2 Preview • Find all solutions:

11. Chapter 2 Preview • Find all solutions:

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

13. 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

14. 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

15. 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]

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

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

18. 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> 0PASS • Interval solutions are [-9, 1) and [5, ∞)

19. 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

20. 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

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

22. Chapter 2 Preview • #20, continued • 16t2-128t+240 < 0 • 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.