Quadratic Problems

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Quadratic Problems

• Babylonians
• And
• Diophantus

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This type of problem appears on a Babylonian
tablet 1700BC
Write this down.
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Like we were doing in false position, lets make
a guess

• 5 x 5 25 sq. units, but we wanted
• xy 16
• How far off are we?
• 25 - 16 9 units of area.
• Error 9 sq. units

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(No Transcript)
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Our solutions for the sides of the rectangle are
the lengths 8 and 2.
Lets try another
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Try this one on our own
In English, I am looking for two numbers whose
product is 45 and whose sum is 18.
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What did you get?
The two numbers whose product is 45 and whose
sum is 18 are 3 and 15.
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Diophantus (200 - 284)

• There are, however, many other types of
  problems considered by Diophantus. (MACTUTOR
  Biography)

Try this in your group.
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Did you get 9 and 1 for y and z ?

Diophantus idea is to plan ahead a bit.
He introduces x to be the difference that we
would soon be adding and subtracting from 5 if
we were to do it the Babylonians way. So he
replaces y by (5 x) and z by (5 - x) yz 9
becomes (5 x)(5 - x) 9
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yz 9 becomes (5 x)(5 - x) 9
25 - x2 9
x2 16
So, X 4
So we get y 9 and z 1 just as Diophantus
tells us to do.
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Whats been the point of the two talks on False
Position?

• To see and understand how early mathematicians
  solved equations
• To experience a style of doing algebra that is
  different from the way we have been taught and
• To wonder at how much early scribes and others
  really understood.

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Thanks for your attention and work !
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How does this system relate to our quadratic
formula?

• Lets consider

Then our first guess is 1/2 b.
Our error will be (1/2 b)2 - c. Take the
square root of that
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So that square root is the amount that we must
add and subtract from 1/2b. This is what we get
for solutions
and
leads to
-gt
Our equation
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OK, now were done!