Lesson 8 - Practice Problems
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1 Lesson 8 - Practice Problems Section 8.1: A Case for the Quadratic Formula 1. For each quadratic equation below, show a graph in the space provided and circle the number and type of solution(s) to that equation. Equation Graph Solutions (Check the Box) a)! x 2 + 2x + 1 = 0 1 real (repeated) b)! x 2 7 = 0 1 real (repeated) c)! 3x 2 4x +10 = 0 1 real (repeated) d)! 3x 2 + 4x = 1 1 real (repeated) e)! 2x 2 + 7= 3x 1 real (repeated) f)! x 2 + 6x 15 = 0 1 real (repeated) j)! 5x 2 = 0 1 real (repeated) 339
2 Section 8.2: Solving Quadratic Equations with the Quadratic Formula 2. Simplify each of the following as much as possible over the real number system. Leave answers in exact form.! a)! 49 =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 121 =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 625 =! b)! 108 =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2 72!=!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 192 =! c)! =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 2 72! 6 =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 192 = ! d)! =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 4 60 =! 4 6 8! e) 49 =!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 121!=!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 625 = 340
3 Quadratic Formula: x = b ± b 2 4ac 2a 3. Solve each quadratic equation by using the Quadratic Formula. Place your given quadratic equation in standard form Identify the coefficients a, b and c Graph to determine number and type of solutions Substitute these values into the quadratic formula Simplify your result completely Mark and label the solutions on your graph a) Solve! 2x 2 2x 4 = 0 (This one is a fill in the blank) a =, b =, c = x = ±! 4 2 x = ± Sketch the graph on a good viewing window (the vertex, vertical intercept and any horizontal intercepts should appear on the screen). Mark and label any real solutions on the graph. x = ± x! =! and x! =! x! =! and x! =! x! = and x! = x! = 2 and x! = 1 Final solution x = 1, 2 Xmin = Xmax = Ymin = Ymax = 341
4 Quadratic Formula: x = b ± b 2 4ac 2a 4. Solve each quadratic equation by using the Quadratic Formula. Place your given quadratic equation in standard form Identify the coefficients a, b and c Graph to determine number and type of solutions Substitute these values into the quadratic formula Simplify your result completely Mark and label the solutions on your graph a)! 2x 2 5x = 4 b)! 4x 2 2x = 6 342
5 Quadratic Formula: x = b ± b 2 4ac 2a 5. Solve each quadratic equation by using the Quadratic Formula. Place your given quadratic equation in standard form Identify the coefficients a, b and c Graph to determine number and type of solutions Substitute these values into the quadratic formula Simplify your result completely Mark and label the solutions on your graph a)! 6x 2 4x = 1 b)! 3x 2 x 4 = 0 343
6 Section 8.3: Complex Numbers 6. Simplify each of the following and write in exact a + bi form. Where possible, write in approximate form with answer to the thousandths place. a) 81 = b) 11 = c)! 8 = d)! 2 49 = e)! = f)! 2 8 = g) h) 4 8 = 6 i) = j) 2 4 4(2)(5) = 4 344
7 Section 8.4: Complex Solutions to Quadratic Equations 7. Solve the quadratic equations in the complex number system by using the Quadratic Formula. Leave your final solution in the complex form, a ± bi. Sketch the graph on a good viewing window (the vertex, vertical intercept and any horizontal intercepts should appear on the screen). Mark and label any real solutions. a) x! + 5x + 17 = 0 b) x! + 2x + 5 = 0 345
8 8. Solve the quadratic equations in the complex number system by using the Quadratic Formula. Leave your final solutions in the complex form, a ± bi. Sketch the graph on a good viewing window (the vertex, vertical intercept and any horizontal intercepts should appear on the screen) and verify that your solutions are complex. a) 4x! = 9 b) 3x! + 4x 7 = 0 346
9 Section 8.5: Combining Solution Methods to Solve Any Quadratic Equation 9. Given the quadratic equation x 2 + 4x 3 = 2, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a ± bi. Clearly identify your solutions in all cases. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept, and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. Solutions:! x 1 =! x 2 = b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 347
10 10. Given the quadratic equation! 2x 2 + x 3= x, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a ± bi. Clearly identify your solutions in all cases. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept, and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. Solutions:! x 1 =! x 2 = b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 348
11 11. Given the quadratic equation! x 2 + 6x + 9 = 0, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a ± bi. Clearly identify your solutions in all cases. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept, and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. Solutions:! x 1 =! x 2 = b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 349
12 12. Given the quadratic equation! x = 0, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a ± bi. Clearly identify your solutions in all cases. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept, and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. Solutions:! x 1 =! x 2 = b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 350
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