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Consider the quadratic equation 2x2 150x+3=0 (i) Compute approximate roots by solving 2x²-150x = 0 (1) (2) (If a root finding method was being implemented to solve equation (1), these approximations might be used to form the initial bracketing interval(s).) (ii) Use the quadratic formula to compute the roots of equation (1). (iii) Fill in the following table using the nominated digits of precision in your calculations. These should be hand/calculator calculations. Show example calculations for 2 digits of precision. Calculations do not have to be shown for the remaining cases. digits of precision 62-4ac √b2-4ac | x2 = (-b-√√b2-4ac)/(2a) 1 2 3 4 5 6 (iv) With the standard quadratic solutions defined as 1 x1 = -b+ √√b2-4ac 2a and 1 x2 2a (-b- it is straightforward to show that x1x2 = ' - √b² - 4ac) which then rearranges to give (3) (4) C x1 and x2 ax2 C а х1 for any general a, b, c. If catastrophic cancellations lead to one of expressions (3) or (4) being inaccurate untrustworthy, it is replaced by the relevant expression in (5). Discuss the conditions (in terms of general a, b, c) under which expression (3) can be trusted and (4) cannot because of catastrophic cancellations. Repeat for the case when (4) can be trusted and (3) cannot. You are not to use the values in equation (1) in your answer, but their relative values might guide you in your analysis. (v) The table in (iii) should show a lot of erroneous calculations of x2 due to catastrophic cancella- tions. Fill in the following table using the nominated digits of precision, to show that equation (5) leads to improved calculations for x2. Again, these should be hand/calculator calculations. digits of precision | x₁ = (-b+√b2-4ac)/(2a) | x2 = ax1 1 2 3 4 5 6

Consider the quadratic equation 2x2 150x+3=0 (i) Compute approximate roots by solving 2x²-150x = 0 (1) (2) (If a root finding method was being implemented to solve equation (1), these approximations might be used to form the initial bracketing interval(s).) (ii) Use the quadratic formula to compute the roots of equation (1). (iii) Fill in the following table using the nominated digits of precision in your calculations. These should be hand/calculator calculations. Show example calculations for 2 digits of precision. Calculations do not have to be shown for the remaining cases. digits of precision 62-4ac √b2-4ac | x2 = (-b-√√b2-4ac)/(2a) 1 2 3 4 5 6 (iv) With the standard quadratic solutions defined as 1 x1 = -b+ √√b2-4ac 2a and 1 x2 2a (-b- it is straightforward to show that x1x2 = ' - √b² - 4ac) which then rearranges to give (3) (4) C x1 and x2 ax2 C а х1 for any general a, b, c. If catastrophic cancellations lead to one of expressions (3) or (4) being inaccurate untrustworthy, it is replaced by the relevant expression in (5). Discuss the conditions (in terms of general a, b, c) under which expression (3) can be trusted and (4) cannot because of catastrophic cancellations. Repeat for the case when (4) can be trusted and (3) cannot. You are not to use the values in equation (1) in your answer, but their relative values might guide you in your analysis. (v) The table in (iii) should show a lot of erroneous calculations of x2 due to catastrophic cancella- tions. Fill in the following table using the nominated digits of precision, to show that equation (5) leads to improved calculations for x2. Again, these should be hand/calculator calculations. digits of precision | x₁ = (-b+√b2-4ac)/(2a) | x2 = ax1 1 2 3 4 5 6

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the quadratic equation 2x2 150x+3=0 (i) Compute approximate roots by solving 2x²-150x = 0 (1) (2) (If a root finding method was being implemented to solve equation (1), these approximations might be used to form the initial bracketing interval(s).) (ii) Use the quadratic formula to compute the roots of equation (1). (iii) Fill in the following table using the nominated digits of precision in your calculations. These should be hand/calculator calculations. Show example calculations for 2 digits of precision. Calculations do not have to be shown for the remaining cases. digits of precision 62-4ac √b2-4ac | x2 = (-b-√√b2-4ac)/(2a) 1 2 3 4 5 6 (iv) With the standard quadratic solutions defined as 1 x1 = -b+ √√b2-4ac 2a and 1 x2 2a (-b- it is straightforward to show that x1x2 = ' - √b² - 4ac) which then rearranges to give (3) (4) C x1 and x2 ax2 C а х1 for any general a, b, c. If catastrophic cancellations lead to one of expressions (3) or (4) being inaccurate untrustworthy, it is replaced by the relevant expression in (5). Discuss the conditions (in terms of general a, b, c) under which expression (3) can be trusted and (4) cannot because of catastrophic cancellations. Repeat for the case when (4) can be trusted and (3) cannot. You are not to use the values in equation (1) in your answer, but their relative values might guide you in your analysis. (v) The table in (iii) should show a lot of erroneous calculations of x2 due to catastrophic cancella- tions. Fill in the following table using the nominated digits of precision, to show that equation (5) leads to improved calculations for x2. Again, these should be hand/calculator calculations. digits of precision | x₁ = (-b+√b2-4ac)/(2a) | x2 = ax1 1 2 3 4 5 6
Transcribed Image Text:Consider the quadratic equation 2x2 150x+3=0 (i) Compute approximate roots by solving 2x²-150x = 0 (1) (2) (If a root finding method was being implemented to solve equation (1), these approximations might be used to form the initial bracketing interval(s).) (ii) Use the quadratic formula to compute the roots of equation (1). (iii) Fill in the following table using the nominated digits of precision in your calculations. These should be hand/calculator calculations. Show example calculations for 2 digits of precision. Calculations do not have to be shown for the remaining cases. digits of precision 62-4ac √b2-4ac | x2 = (-b-√√b2-4ac)/(2a) 1 2 3 4 5 6 (iv) With the standard quadratic solutions defined as 1 x1 = -b+ √√b2-4ac 2a and 1 x2 2a (-b- it is straightforward to show that x1x2 = ' - √b² - 4ac) which then rearranges to give (3) (4) C x1 and x2 ax2 C а х1 for any general a, b, c. If catastrophic cancellations lead to one of expressions (3) or (4) being inaccurate untrustworthy, it is replaced by the relevant expression in (5). Discuss the conditions (in terms of general a, b, c) under which expression (3) can be trusted and (4) cannot because of catastrophic cancellations. Repeat for the case when (4) can be trusted and (3) cannot. You are not to use the values in equation (1) in your answer, but their relative values might guide you in your analysis. (v) The table in (iii) should show a lot of erroneous calculations of x2 due to catastrophic cancella- tions. Fill in the following table using the nominated digits of precision, to show that equation (5) leads to improved calculations for x2. Again, these should be hand/calculator calculations. digits of precision | x₁ = (-b+√b2-4ac)/(2a) | x2 = ax1 1 2 3 4 5 6
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