math project

Running head : QUADRATIC EQUATION

Early Forms of Solution to Quadratic Equation

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Quadratic Equation

It is claimed that the Babylonians were the first people to solve a quadratic equation . But during the times of the ancient Babylonians there was no such thing as quadratic equation but they have problems that are solved using the approach of quadratic equation (O 'Connor E F Robertson , 1996 . This method is very much used for finding the sides of a rectangle when only the difference between the sides and

the area of the rectangle are given . To illustrate how the Babylonians method works , consider a rectangle with its length d greater than the width and an area A . the solutions to this problem are as follows

Let L be the length and W be the width of the rectangle . Using the conditions given

L W d (1

A LW (2

Substituting L to (2 ) and collecting terms to the left side of the equation , we arrive at

W2 Wd-A 0

Using the quadratic formula and rejecting the negative root , we have

W (-d (d2 4A /2

L W d (d (d2 4A /2

The Babylonians also arrive at this answer using a series of steps First , they take half the difference between the sides , d /2 . Then they square the result , giving d2 /4 . They add this to the area and take the square root of the sum , arriving at (d2 /4 A ) or (d2 4A /2 . They subtract half the difference to the result to find the width and add half the difference to the result to get the length . The answers are

W (-d (d2 4A /2

L W d (d (d2 4A /2

which are the same answer we get using quadratic formula

Aside from the Babylonians , the Arabs also find a way of solving quadratic equations . But unlike the Babylonians , who solve for the sides of the rectangle , they use the sides and areas of squares and rectangles to solve a quadratic equation . The mathematician who introduced this approach is al-Khwarizmi . For example , to solve the root of the quadratic equation x2 20x 300 , consider a square having side of length x . Putting four 5 by x rectangles on the square , we arrive at a figure below , having a By adding 5 by 5 squares on each side , we will have a square that looks like the figures below

The area of the resulting square is 300 25 25 25 25 or 400 . The side of the bigger square is ? 400 or 20 . From the figure , we could see that this side is 10 longer than x , and so we have x 10 . This method is similar to the modern method of completing the square

Exercises

A rectangular plate is 5 inches longer than it is wide and has an area of 24 square inches . Use the Babylonian method to find the length and the width , indicating each step as you go

Here 's a historical quadratic...