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# Algebra Homework 3 Solutions

Problem 5: Use four-digit rounding arithmetic and the formulas to find the most accurate
approximations to the roots of the following quadratic equations. Compute the
relative error.

a) b) 1.002x2 + 11.01x+ 0.01265 = 0.

Solution : The quadratic formula states that the roots of ax2 + bx + c = 0 are a) The roots of are approximately

x1 = 92.24457962731231,
x2 = 0.00542037268770.

We use four- digit rounding arithmetic to find approximations to the roots. We find the
first root: which has the following relative error: has the following relative error: We obtained a very large relative error, since the calculation for involved the subtraction
of nearly equal numbers . In order to get a more accurate approximation to , we need
to use an alternate quadratic formula , namely Using four- digit rounding arithmetic, we obtain: which has the following relative error: b) The roots of 1.002x2 + 11.01x+ 0.01265 = 0 are approximately

x1 = −0.00114907565991,
x2 = −10.98687487643590.

We use four-digit rounding arithmetic to find approximations to the roots.
If we use the generic quadratic formula for the calculation of , we will encounter the
subtraction of nearly equal numbers (you may check). Therefore, we use the alternate
quadratic formula to find ; which has the following relative error: We find the second root using the generic quadratic formula: which has the following relative error: Similar Problem
The roots of 1.002x2 − 11.01x+ 0.01265 = 0 are approximately

x1 = 10.98687487643590,
x2 = 0.00114907565991.

We use four-digit rounding arithmetic to find approximations to the roots. We find the
first root: which has the following relative error: If we use the generic quadratic formula for the calculation of , we will encounter the
subtraction of nearly equal numbers . Therefore, we use the alternate quadratic formula
to find : which has the following relative error: Problem 6: Suppose that fl(y) is a k-digit rounding approximation to y. Show that (Hint: If then If then )

Solution : We have to look at two cases separately.

Case :   since so  since by assumption Case :  Note that and For example,  since so  Prev Next