**Sheri's work on the quadratic equation x ^{2}
- x - 1 = 0**

Don had Sheri keep track of the number she put **in**
for x, and the number she got **out**. Her aim was to get 0, of course. Now
Sheri made a bunch of mistakes here. She left out the negative on ^{-}1
and she didn't subtract 1 all the time. Don found these mistakes and Sheri
corrected them. Don and Sheri got into changing 1/3 to a decimal, by dividing 1
by 3, which she admitted to not knowing too much about. But she's learning. They
also got into negative exponents **again**, because the calculator gave an
answer ^{-}7.6 E ^{-}5 when she put 1.618 in for x.

After a chaotic beginning, Don asked Sheri to organize her work:

x^{2}
- x - 1 = 0

in out

2 1

1
^{-}1

Sheri wrote, " Our answer is
between **1** and **2**,
because when we put 2 **in** we get 1 **out**. When we put 1 **in** we
get ^{-}1
**out** and 0 is between 1 and ^{-}1".

in out

1.6
^{-}0.04

1.7 0.19

1.5
^{-}.25

1.8 .44

"Our answer is between **1.****6**
and **1.****7**".

in out

1.62 0.0044

1.61
^{-}0.0179

"Our answer is between **1.****61**
and **1****.****62**
because 1.62
**in** gives .0044
**out**, and positive, and 1.61
**in **gives ^{-}0.0179,
**out**, and negative. Our answer is closer to 1.62
".

in out

1.618
^{-}7.6
E ^{-}5
= ^{-}.000076

1.619 0.002161

1.6188 .00171344

1.6182 not finished

(The first output from the
calculator above, ^{-}7.6
E ^{-}5,
means ^{-}7.6
x10^{-}^{5}.
Don and Sheri worked on negative exponents and Sheri finally wrote that as ^{-}.000076
).

At this point Don wrote the first two of Sheri's results below, then Sheri wrote the rest.

Too big
-> **2**, **1****.****7**,
**1****.****62**,
**1****.****619**
,

Too small
->**1**, **1****.****6**,
**1****.****61**,
**1****.****618**
,

Notice that these numbers are honing
in on the answer to the nearest whole number, the nearest tenth, the nearest
hundredth, and so on. These are the beginnings of two ** infinite
sequences**, both
sequences are approaching just **one** of the two solutions of the quadratic
equation x^{2}
- x - 1 = 0. Sheri and Don talked about this.

Tune in for more of this work by
Sheri, **if** she continues.

Sheri did continue; after many calculations, she came up with an approximation to the answer, with 7 decimal places:

Too big
-> **2,** **1****.****7****,**
**1****.****62****,**
**1****.****619,**
**1****.****6181,
1****.****61804,
1****.****618034,
1****.****6180340,...**

Too small
->**1****,**
**1****.****6****,**
**1****.****61****,
1****.****618****,**
**1****.****6180****,
1****.****61803,**
**1****.****618033,
****1****.****6180339,...**

"And our answer is closer to **1****.****6180340****
!**"

At this point Don had Sheri graphed these two sequences. The problem was that the differences get very small very quickly. After plotting 3 points, the numbers were so close together they plotted one more.

Don showed Sheri the book "The
Penguin Dictionary of Curious and Interesting Numbers" by David Wells. She
found this number in the book out to 50 decimal places! Don told Sheri that this
is a very important number in mathematics and life called "The Golden
Mean" or "The Divine Proportion". Don got out the book "The
Divine Proportion" by H. E. Huntley which is all about this number.
The Parthenon in Greece was built in the dimensions 1x**1****.****6180340**,**
**and the most esthetically pleasing
rectangle was found to be this one. He told her it was an irrational number (the
square root of a number NOT a perfect square, and couldn't be written as a
fraction with whole numbers and was NOT an infinite repeating decimal). They
found it written in Wells as .
The next question to Sheri will be, if this is one solution of our quadratic
equation, what is the second number?

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in Mathematics

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