What is the significance of the other answer in the quadratic formula?

[WardenOfTheMint]

I know how it works, used to find the unknown variable...been using it since Algebra 2...

What do the 'two' answers represent. I know one of them always seems reasonable, but the other one is whack (but it probably has some significance).

Can someone give me an example of the significance of 'that other answer.' Use any example.

 

[Borek]

Imagine you throw a stone at the 45 deg angle in the huge greenhouse. It will make two holes in the roof. Their positions are both solutions of quadratic formula.

 

[Redbelly98]

Warden,

Sometimes the two solutions are both reasonable. It really depends on the specific problem or application.

If you can give an example of where you saw this come up, we might be able to help with why one answer is unphysical for your particular example.

 

[lukas86]

Haha, I like your explanation on that one Borek

 

[GTrax]

OK - so you know a quadratic equation has the characteristic that the variable contains at least a squared term, and possibly other lower order terms. You know also it has two roots (solutions) You are OK with that too, except that when you have applied it, you find often only one is realistic, and the other has to be ignored -- thrown away as absurd and meaningless.

This is the difference between pure and applied math. The quadratic is good for all values of its variable, but some range of these may not be in a range suitable for use in modelling the real world. This is all about setting the valid boundaries at the outset.

Consider the example of using a Fourier series of sines and cosines to accurately represent a periodic electrical signal waveform. The series is good for variable values -infinity through to +infinity. The variable is 2*pi*frequency. The whole concept of "negative" frequency is meaningless! So we set the limits to this modelling. No solutions involving negative frequency allowed. It happens often!

 

[vociferous]

A quadratic can have one solution or it can have no real solution (although, I suppose you could say that it has "two equal solutions" or "two complex solutions".

If you graph the equation, you can see why it might have one solution, two solutions, or no solutions/complex solution.

 

[GTrax]

Ahh yes..

You mean something like (x-2)(x-2)=0. I guess I was just trying to illustrate the point, and I did not pay too much attention to being complete and precise.

For WardenOfTheMint, let us be clear that quadratics (or higher order even) equations applied to model any real phenomena involving parabolic (or other order) phenomena can have solutions real, or none, or some of either kind, including duplicates. Some will be meaningless and inappropriate.

For pure math, there is no problem in including them all. For applied math, we have to also apply some common sense. Even so, I am sometimes struck by how far some mathematical abstractions and transforms that appear completely meaningless as applied to modeling real situations, can still be involved in arriving at an accurate applied math expression for real physical stuff.

 

[vociferous]

Yes, it is very important to understand how the mathematical model is applied to the physical problem.

For instance, if you throw a from the edge of a cliff, it is only going to hit the ground once, but if you represent it as a quadratic of time versus height, you will have two mathematical solutions, but since one is going to represent where the rock was before you threw it, you can safely ignore it.