Unit 3: HW3.5 Sum and Product

Transcription

1 Unit 3: HW3.5 Sum and Product Without solving, find the sum and product of the roots of each equation. 1. x 2 8x + 7 = x + 5 = x x + 4 = -3x x 2 = 5x x 2 2x x2 3x 2 Write a quadratic equation whose roots r 1 and r 2 have the following sums and products. 7. sum = 5; product = 6 8. r 1 + r 2 = -7; r 1 r 2 = 5 9. sum = -11; product = r 1 + r 2 = -1/2 ; r 1 r 2 = sum = -4; product = 2/3 12. r 1 + r 2 = 3/8 ; r 1 r 2 = -1 Write the quadratic equation in standard form with the following roots. 13. {-2, 7} 14. {-1/4, 12} 1Unit 3 Quadratic Functions 2

2 15. {3 5, 3 5} 16. {3i, -3i} 17. {2 i 2, 2 i 2} 18. {3 4i,3 4i} In each case, one root r 1 of a quadratic equation is given. Find the other root and the missing coefficient. 19. x 2 5x + c = 0; r 1 = x 2 + bx 7 = 0; r 1 = x 2 6x + c = 0; r 1 = x 2 + bx + 8 = 0; r 1 = 2 Answers 1. 8, , /3, 4/ /2, -1/5 5. 2/15, 4/ , x 2-5x+6=0 8. x 2 +7x+5= 0 9. x 2 +11x - 10= x 2 +x+4= x 2 +12x+2= x 2-3x-8=0 13. x 2-5x-14= x 2-47x-12=0 15. x 2-6x+4=0 16. x 2 +9=0 17. x 2-4x+6=0 18. x 2-6x+25=0 19. r 2 =3, c=6 20. r 2 =7, b= r 2 =4, b= r 2 =2, b= - 8 1Unit 3 Quadratic Functions 3

3 Unit 3: HW3.6 Solving Quadratics Solve the equation using the method listed. Verify the solutions by finding the sum and the product of the roots. 1. Solve by completing the square: x 2 + 8x + 20 = 0 Check: 2. Solve by factoring: 7x 2 = 2x + 5 Check: 3. Solve using the quadratic formula: 1 2 x 2 4 2x Check: 1Unit 3 Quadratic Functions 6

4 Calculate the discriminant and then describe the nature of the roots as one of the following: a. real, rational, unequal b. real, irrational, unequal c. real, rational, equal d. imaginary 4. x 2 + x + 1 = 0 5. x 2 + 2x + 1 = x 2 + 3x + 2 = 0 7. x 2 2x 2 = x 2 = 5 9. x 2 9 = For what value of k are the roots of 2x 2 8x + k = 0 equal? 11. If the roots of the equation ax 2 + bx + c = 0 are imaginary, the graph of y = ax 2 + bx + c will a. intersect the x-axis at two points b. not intersect the y-axis c. not intersect the x-axis d. be tangent to the x-axis 12. Which is true of the graph of y = x 2 + 4x + 6? a. It intersects the x-axis at two points b. It does not intersect the y-axis c. It does not intersect the x-axis d. It is tangent to the x-axis 13. A quadratic equation having the roots (2 3i) and (2 + 3i) is a. x 2 + 4x + 13 = 0 b. x 2 + 4x 9 = 0 c. x 2 4x + 13 = 0 d. x 2 4x + 9 = 0 1Unit 3 Quadratic Functions 7

5 Unit 3: HW3.11 Solving Polynomials Solve the equation. Check using your calculator 1. (x 2 + 4x + 4)(x 2 9) = 0 2. x 5 10x 3 + 9x = 0 3. x 4 = 29x x 3 + 2x 2 = 8x Use the box method to find the roots. 5. f(x) = x 4 + 6x 3 + 9x 2 0

6 6. g(x) = (x + 2) 3 (x 4) 2 Solve the equation. Check using sum and product. 7. 2x 2 = 6x 8 Check: 8. 3x 2 + 9x + 1 = 0 Check: 9. x 2 + 4x 5 0 Check: Graph: 10. y < 2x 2 + 8x 3 1

7 Unit 3: HW3.12 Solving Polynomials Given the graph below, estimate the real roots of the function. Use your calculator to check your estimates. Then write the function as a product of the factors. 1. f(x) = x 3 13x g(x) = x 3 4x 2 x +4 f(x) = g(x) = 3. h(x) = x 4 9x x 2 4. k(x) = x 5 7x x x 2 24x h(x) = k(x) = Solve the following algebraically x 3 3 = 0 4

8 6. 32 = 2x 4 7. x 3 = 4x 2 4x Graph: 8. y x 2 < 4x + 4 Solve: 9. x = 0 5

9 Day 13 Unit Review Graph the function. Label the vertex and the axis of symmetry 1. y = x y = x 2 2x + 1 Tell whether the function has a minimum or a maximum value. Then find the value. 3. y x y x 2 5x 6 Graph the function. Label the vertex and the axis of symmetry. Also label the x-intercepts. 5. y 2 x 1 2 y x 2 x 2 6. Graph the following quadratic inequalities. 7. y < - x 2 2x y 4x 2 x - 7 6

10 Write the quadratic function in standard form. 9. y 2(x 3)(x 1) 10. y 5(x 2) A child at a swimming pool jumps off a 12 foot platform into the pool. The childs height in feet above the water, is modeled by h(t) = -16t , where t is the time in seconds after the child jumps. a. How long will it take for the child to reach the water? b. What is the maximum height the child reaches? Solve the following equations. 12. x 2 + 5x = c 2 6c 18 = x 2-24 = x = 0 7

11 Solve the following equations using completing the square technique x 2 = 18x u 2 8u + 15 = 0 Solve the following equations using the quadratic formula x 2-7 = 3x 19. x 2 2 x 1 Rewrite the following quadratic equations in vertex form and state the vertex. 20. y = x 2 + 4x y = x 2 5x + 12 Solve the following quadratic inequatlities and graph the solution. 22. x 2 7x + 12 > x 2 + 2x 3 8

12 Solve the following equations algebraically x x 4 3x = 0 State the real solutions of the following polynomials. 26. f(x) = 4x 5 8x 4-5x x 2 + x g(x) = 4x 4 17x Evaluate the discriminant of the given equation. Then describe the nature of the roots as 1: 2 real rational; 2: 2 real irrational; 3: 1 real rational; 4: 2 imaginary 28. 2x 2 7x + 1 = x 2 4x + 1 = Find the value of c which makes the graph y = x 2 4x + c tangent to the x-axis. 9

13 31. Find the smallest value of k which will make the solutions of the quadratic equation imaginary. 2x 2 + 4x + k = 0 Without solving, find the sum and product of the roots of each equation x 2 4x + 5 = x + 7 = 5x 2 Write a quadratic equation whose roots r 1 and r 2 have the following sums and products. 34. sum = 3; product = r 1 + r 2 = -4; r 1 r 2 = 9 Write the quadratic equation in standard form with the following roots. 36. {2 7, 2 7} 37. {8 + 7i, 8 7i} In each case, one root r 1 of a quadratic equation is given. Find the other root and the missing coefficient. 38. x 2 5x + c = 0; r 1 = x 2 + bx 8 = 0; r 1 = 2 4Unit 3 Quadratic Functions 0

14 11R Unit 3 Day 13 Review Ans Key 1. Graph: Vertex (0, -1) AOS: x = 0 2. Graph: Vertex (1, 0) AOS: x = 1 3. maximum: (0, 1) 4. minimum: (2.5, -.25) 5. Graph: Vertex (1, 0) AOS: x = 1 x-intercept (1, 0) 6. Graph: Vertex (0, 4) AOS: x = 0 x-intercepts (-2, 0), (2, 0) y (x 5 2 ) Vertex 2, (,3) U (4, ) 23. [-3, 1] 24. { 5 8 5i 3 8, 5 8 5i { 1, 1, 2, 2 }, 5 4 } 26. { 1, 1 2, 1,1,2} (check on calc) { 2, 1 2, 1 2,2} real irrational roots real rational root 30. c = k > sum = 2, product = sum = -2, product = y = 2x 2 + 4x y = 5x 2-20x a seconds b. 12 feet 12. {-7, 2} 13. {-1.5, 3} 14. { 2 2, 2 2} 15. {3i, -3i} 16. {3 2 2,3 2 2} 17. {3, 5} 18. { , } 19. {1 i,1 i} 20. y = (x + 2) 2 12 Vertex (-2, -12) 34. x 2-3x 8 = x 2 + 4x + 9 = x 2-4x 3 = x 2-16x = r 2 = 8 ; c = r 2 = -4 ; b = 2 4Unit 3 Quadratic Functions 1