Trial and Improvement Objectives C Grade Form and
Trial and Improvement Objectives: C Grade Form and solve equations such as x 2 + x = 12 using trial and improvement
Trial and Improvement Estimate the square root of the following numbers: a. b. c. d. e. f. 17 30 47 68 110 83 4. 123105626 5. 5 5. 477225575 6. 9 6. 8556546 8. 246211251 10. 5 10. 48808848 9. 110433579 Now check your answers on a calculator
Trial and Improvement 85 Find the positive solution to the equation x 2 - x = 60 give your answer to 1 decimal place If we consider this drawing a graph y = 60 we know the solution can be found by drawing the line y = 60 and y = x 2 – x, finding the value of x at the point of intersection. because of the scale of the graph we have to use we cannot find the value of x to 1 d. p. but we can see it is between 8 and 9. 80 75 y = x 2 - x 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 -5 -4 -3 -2 -1 0 1 2 3 46, 9999999 57, 9999999 68, 9999999 -5
Trial and Improvement 85 Trial and improvement is where we try values of x in the equation and try to get as close to the given value for y as possible 2 – x = 60 Now the solution is Try x = we 8 knowxthat 8. 25 x =we 8 istrytoo low 64 between – 8 = 568. 2 and 8. 3 80 75 70 65 60 55 50 Try x = 9 81 – 9 = 72 x = 9 is too low Try x = 8. 5 72. 25 – 8. 5 = 63. 75 x = 8. 5 is too high Try x = 8. 3 68. 89 – 8. 3 = 60. 59 x = 8. 3 is too high Try x = 8. 2 67. 42 – 8. 2 = 59. 04 x = 8. 2 is too low 45 40 35 30 25 20 Now even the expanding 15 10 graph is not big enough 5 for the level 0 of required -5 -4 -3 -2 -1 0 accuracy 1 2 3 46, 9999999 57, 9999999 68, 9999999 -5
Trial and Improvement Trial and improvement is where we try values of x in the equation and try to get as close to the given value for y as possible Now we know that the solution is between 8. 2 and 8. 3 we try 8. 25 Try x = 8. 25 68. 0625 – 8. 25 = 59. 8125 x = 8. 25 is too low We need the value of x to 1 d. p. We know the solution is now between 8. 25 and 8. 3. Try x = 8. 3 Nowbeeven the expanding 8. 25 and 8. 3 would rounded to 8. 3 68. 89 – Any 8. 3 =value 60. 59 between x = 8. 3 is too high graph is not big enough to 1 d. p Try x = 8. 2 for the level Therefore to 1 d. p x = 8. 3 67. 42 – 8. 2 = 63. 75 x = 8. 2 is too low of accuracy required
Trial and Improvement Find the value of x to 1. d. p to solve this equation: x 3 + x = 12 We can now do this as a table: Trial Value of x 3 2 2. 1 2. 2 2. 15 2. 14 x 3 x x 3 -x 27 3 30 8 2 10 9. 261 2. 1 11. 36 10. 65 2. 2 12. 85 9. 938 2. 15 12. 09 9. 8 2. 14 11. 94 Comment Too High Too Low Because 2. 15 is too high and any number less than 2. 15 would be rounded to 2. 1 to 1 d. p. Finding the answer when x = 2. 14 proves that this is correct.
Now do these: Trial and Improvement Find the positive solutions to 1 decimal place 1. x 2 + 2 x = 63 2. x 2 - 2 x = 675 3. x 3 + 2 x = 520 4. x 5 + x = 33 768 5. x 2 - 7 x = 368 6. x - x 3 = -336
Trial and Improvement Worksheet x 2 + 2 x = 63 Trial Value of x x 2 - 2 x = 675 Trial Value of x x 3 + 2 x = 520 Trial Value of x x 2 x 3 2 x 2 x 2 x x 2 + 2 x Comment x 5 + x = 33 768 Trial Value of x x 5 + x Comment x 2 - 2 x Comment x 2 - 7 x = 368 Trial Value of x x 2 7 x x 2 - 7 x Comment x 3 + 2 x Comment x - x 3 = -336 Trial Value of x x x 3 x - x 3 Comment
Trial and Improvement x 2 + 2 x = 63 Trial Value of x 5 6 7 x=7 63 x 2 25 36 49 x 2 - 2 x = 675 Trial Value of x 20 30 28 26 27 x = 27 675 x 2 400 900 784 676 729 2 x 10 12 14 2 x 40 60 56 52 54 x 520 = 8. 0 x 3 + 2 x = 520 x 3 2 x Trial Value of x 8 512 16 7 343 14 7. 5 421. 9 15 7. 9 493 15. 8 7. 95 502. 5 15. 9 7. 96 504. 4 15. 92 x 2 + 2 x Comment 35 Too Low 48 Too Low 63 x 2 - 2 x 360 840 728 624 675 x 3 + 2 x 528 357 436. 9 508. 8 518. 4 520. 3 Comment Too Low Too High Too Low Comment Too High Too Low Too High x 5 + x = 33 768 Trial Value of x 8 9 8. 1 8. 05 8. 02 8. 04 33768 x = 8. 0 x 5 x 32768 16 59049 18 34868 16. 2 33805 16. 1 33180 16. 04 33595 16. 08 x 5 + x Comment 32784 Too Low 59067 Too High 34884 Too High 33821 Too High 33196 Too Low 33612 Too Low x 2 - 7 x = 368 Trial Value of x 25 22 23 368 x = 23 x 2 7 x 625 175 484 154 529 161 x 2 - 7 x Comment 450 Too High 330 Too Low 368 x = 7. 3 x - x 3 = -336 x x 3 Trial Value of x 7 49 343 8 64 512 7. 5 56. 25 421. 9 7. 4 54. 76 405. 2 7. 3 53. 29 389 x - x 3 Comment -294 Too High -448 Too Low -366 Too Low -350 Too Low -336
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