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Name: Lanette
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What are some real world applications of completing the square?

Lanette, One fairly simple example is a mass hanging on a spring. An easy equation to solve is E=(1/2)mv^2+(1/2)kx^2. The solution is a sine wave: x=Asin(wt), v=wAcos(wt). Things are not so simple when gravity is added: E=(1/2)mv^2+(1/2)kx^2+mgx. Complete the square for the x-terms: E=(1/2)mv^2+(1/2)k{x^2+(2mg/k)x+(mg/k)^2}-(1/2)k(mg/k)^2


Now, let x'=x+mg/k. Changing the position by a constant value will not affect velocity: v'=v. As we can see, the energy is shifted: E'=E+(m^2g^2)/2k. The problem has been reduced to its original form: E'=(1/2)mv'^2+(1/2)kx'^2. Solve for x', and then convert back to x=x'-mg/k.

This is not the only application of completing the squares, but it is the first one that comes to mind.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

There are so many examples of the use of "completing the square" -- too many to give just one "real world" example. That misses the point. In case it has not been explained to you to your satisfaction, here is the "reason" for this particular algebraic operation. If you have an algebraic expression that can be expressed as: [EXPRESSION]^2 = [A SIMPLER EXPRESSION] "completing the square" lets you simplify the LEFT SIDE OF THE] and the right hand side of the equation, hopefully is simpler. You can take the SQUARE ROOT OF BOTH SIDES OF THE EQUATION and hopefully the result can be made simpler.

So "completing the square" is only a method for simplifying a computation. It is a "formula" for rearranging an equation. It is not a Huge new concept in mathematics.

Vince Calder

Completing the square of an algebraic polynomial is just a method for simplifying the expression. There is no fundamental secret or magic about carrying out this operations. If by carrying out operations to give an expression of the form: [EXPRESSION #1]^2 = [SOME NUMBER] OR [OTHER EXPRESSION #2] then taking the square root of both sides of the equal sign gives [EXPRESSION #1] = +/- [THE RIGHT HAND SIDE]. There is no magic, it is just a manipulation that can simplify an equation.

Vince Calder

I have to admit, I do not need to complete any squares on a daily basis. However, when I built myself a hover craft some time ago, I did need to complete several squares to figure out exactly where a certain reference line went.

I knew I wanted to round off my corners, but not with circles. I knew that the portions of ellipses that I did want could be drawn with a string wrapped around two fixed points. And I knew how long I wished the major and minor axis of the ellipse portion to be. Unfortunately, I did not know the length of the string I would need, nor how far apart the two fixed points should be. Only by applying both geometry, and the principle of completing the square (several times), was I able to determine exactly where my reference line went. The end result was a much neater and smoother curve than anything I could have achieved through guesswork.

Ryan Belscamper

"Real world applications" of completing the square kind of misses the point of carrying out that process. "Completing the square" is one of a large number of algebraic manipulations that one uses to convert an algebraic expression into a simpler form. And the whole subject of algebra has to do with solving algebraic equations. So like the "quadratic" formula, and various other techniques for factoring and simplifying algebraic expressions, the motive is to obtain an expression that is easy to solve an algebraic equation with the minimum amount of pain and discomfort.

Vince Calder

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