So I was just thinking about some algebra and wanted to see if anyone else had heard much about the ancient methods by which people used to solve quadratic equations. For those unfamiliar with the term, a quadratic equation is one that has the form Ax^2 + Bx + c = 0 where A, B, and c are numbers while X is a variable. In modern times we can solve these equations with the equation (-B +/- sqrt(c^2 - 4*A*c))/(2*A) however in the past they were often solved visually using a technique called Completing the Square. As an example we can consider the equation [x^2 + 4x = 5]. To complete the square for this technique we first consider x^2 to represent a square whose sides measure x by x. To this box we are adding a rectangle whose sides measure 4 by x. If we were to cut this rectangle in half we would form two smaller, equally sized rectangles of measure 2 by x. We next take these rectangles and place them against two adjacent sides of the box with sides of x by x, taking care to ensure that the side with x is against the box for each of the rectangles. The shape formed by this is an L with Height and Length equal to 2+x, which was necessary to complete the square. To this L shape we will now add a square whose measurements are 2 by 2, this, all together, will form a square with sides of equal length. We account for the addition of this box by adding it to the original equation, yielding [x^2 + 4x + 4 = 5 + 4] => [x^2 + 4x + 4 = 9]. This is the trick behind the technique, by creating a new square with a larger area we gain the ability to determine the overall length of the square. Since the length of the sides of a square is equal to the square root of its volume we take the square root of 9 and get 3. If the overall length of the completed square is equal to 3 and the length of the sides of the square are equal to x+2 we can set the equations equal to each other and solve like so: [x+2=3 -> x=1].
This is a very useful and intuitive approach, however it fails to help us account for negative solutions to the equation. When graphed equations of the form Ax^2 + Bx + c create a parabola. In our example earlier we can think of the number 5 as being representative of a horizontal line parallel with the x axis possessing a height of 5. The solutions for our equation are the two points at which the parabola formed by x^2 + 4x intersect with the line. Additionally, if we consider the equation again we can see that there is another solution made especially clear by re-writing the equation as such: [(x * x)+(4 * x) = 5]. If we plug -5 into our equation we get [(-5*-5)+(4*-5)=5] which can also be written as [(-1*-1)*(5*5)-(4*5)=5]=>[(5*5)-(4*5)=5]. When written out like this it is fairly clear to see that -5 is an answer, but it wasn't until surprisingly recently that negative answers were actually considered valid solutions because the traditional approach to completing the square would require one to construct a square of sides with negative length, something that does not seem to exist in reality. I am not entirely sure I agree with this approach and was wondering if anyone has any thoughts on ways we could construct a geometric representation of the negative solutions in a way similar to that used above. I have come up with a few ideas, but am not sure of how good they are yet, however one way I have constructed that works for some functions like the example above requires one to first solve for the positive solution and then use that to construct two squares, one of which equal in size to the (initial area)^2 and the other equal to the (initial area - 1)^2 and subtract one from the other, which does yield the correct answer, but feels highly limited and nowhere near as general as completing the square. If anyone has any ideas or even just thoughts on the topic I'd love to hear them!
The ancient method of completing the square
So I was doing some more work on this and realized that it is actually not very difficult to derive the quadratic equation from completing the square! Furthermore, there is actually a unique symmetry present in the quadratic equation and the representation of a parabola in which both solutions are equidistant from a mid-point defined as -B/2A, now to just see how to best represent it.
