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Quadratic equation question 
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#1
Dec2613, 04:23 AM

P: 38

consider the equation:
ax^2+bx=c imagine that the constant c is divided into to parts: c=Z+Y where ax^2= Z and bx=Y so basically could we say c is part area and part length?? Sorry if the question is not explained better but that's the best way I came up with to describe it. 


#2
Dec2613, 11:58 AM

Mentor
P: 12,081

If x is a length, then a, b and c have to have different units  as an example, a could be a number, b could be a length, and c could be an area: $$2x^2 + (3m)x = 5m^2$$ has the solution $$x=1m$$ 


#3
Dec2713, 03:02 AM

P: 38

Well i'm considering a and b to be just scalars



#4
Dec2713, 05:57 AM

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P: 5,496

Quadratic equation question
You are needing a practical example? The cost of putting in a square garden bed comprises 3 expenses: → a fixed cost to install a nearby tap, → a cost per meter for a brick border around the bed, and → a cost per m^{2} to provide a fixed depth of good soil. Neither of the coefficients of the variable is a scalar (without units). If you are still unconvinced, please explain where/why you believe you would ever want to sum an area and a length? 


#5
Dec2713, 06:28 AM

P: 38

Well I'lll try to explain it with this picture



#6
Dec2713, 07:00 AM

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P: 5,496

Maybe I can set it to show your pic as an inline display? Let's see ..... Ah! Now that I can see! It reveals that you are not wanting to sum an area and a length, just as I foretold. You are actually wanting to sum "pieces" with "pieces". Just as one can sum oranges with oranges, it is equally valid to sum pieces with pieces. The units of your coefficients will not be dimensionless, as we already indicated. For your example, one coefficient will have units of pieces/metre^{2}, the other will have units of pieces/metre. 


#7
Dec2713, 07:22 AM

P: 38

This is the only way I could explain it, I express myself better with a pen and a paper, it's a shame



#8
Dec2713, 07:48 AM

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#9
Dec2713, 10:04 AM

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P: 1,991

You can think of it with square bricks. ax^{2} is a squares of bricks, each side of length a, then bx is a line of b sets of bricks each of length x, or alternatively a rectangle of sides a and x, total number of bricks equals c. You can even work out solution for x whole numbers, even general solution, by moving bricks around. You can envisage a similar thing for cubic equations.
People did sometimes think about them this way. As it is hard to think about quartic and higher equations this way there were at one time many people who said  you mustn't! Such things are 'not physical', not practical, and abstract etc. No one would say quite that now  but you may well happen to hear some of the same sort of attitude expressed, in tones of great self confidence, by marginally or nonmathematical scientists still today. 


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