# 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) 1.002x^{2} + 11.01x+ 0.01265 = 0.

Solution : The quadratic formula states that the roots of ax^{2} + bx + c = 0 are

a) The roots of are approximately

x_{1} = 92.24457962731231,

x_{2} = 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 1.002x^{2} + 11.01x+ 0.01265 = 0 are approximately

x_{1} = −0.00114907565991,

x_{2} = −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 1.002x^{2} − 11.01x+ 0.01265 = 0 are approximately

x_{1} = 10.98687487643590,

x_{2} = 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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