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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

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