Wikipedia:Reference desk/Archives/Mathematics/2009 February 8

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February 8

Linear Forms of Quadratic Equations

Yesterday I was reading somethings about absolute values and suddenly I thought you could express quadratic functions with absolute valued linear functions! This is how you transform quadratic functions into absolute valued linear functions:
t+│sx+k│=ax^2+bx+c=0
s=±√a
k=b/(2sx)
t=±√(k^2-c)
ax^2+bx+c=s^2x^2+2ksx+k^2-t^2=t+│sx+k│=0
and this is how you transform absolute valued linear functions into quadratic functions:
ax^2+bx+c=t+│sx+k│=0
│sx+k│=-t=√((sx+k)^2)
(sx+k)^2=t^2=s^2x^2+2ksx+k^2
s^2x^2+2ksx+k^2-t^2= t+│sx+k│=ax2+bx+c=0
I want to ask has anyone discovered these forms of quadratic functions yet and did I do anything wrong in my calculations above?The Successor of Physics 03:24, 8 February 2009 (UTC)

Your use of the equal sign is wrong ... you mean that s^2x^2+2ksx+k^2-t^2=0 implies that t+│sx+k│=0, don't you (but it rather implies that |t|-|sx+k|=0)? Anyway, the latter equation is not at all linear in x because k=b/(2sx). Icek (talk) 08:10, 8 February 2009 (UTC)

Oh! the implies sign must have changed into an equal sign when I transferred it from Microsoft word to wikipedia! sorry about that. also, I don't really mean linear, but I only mean its form is linear. The formula for finding roots to these equations are a lot simpler than the quadratic formula; x=(±t-k)/s, and for quadratic functions with a non unit coefficient for x², my method is simpler than factorizing.The Successor of Physics 10:07, 11 February 2009 (UTC)

Vector fields and surface normals

If a vector field B(x) is parallel to the normals of a family of surfaces of f(x)=constant, what do we know about it?

I'm eventually aiming at proving B.curlB=0, but initially I'm just looking for a starting point here. I'm not even totally sure about the 'family of surfaces' - would it be as the constant varies or as some paramater in f(x) varies?

Cheers, Spamalert101 (talk) 14:40, 8 February 2009 (UTC)Spamalert

If ƒ is continuous then the set of all values of x for which ƒ(x) is equal to a specified constant is a surface. If you change the constant you get a different surface. (Maybe I'll come back to your other question.) Michael Hardy (talk) 15:19, 8 February 2009 (UTC)

Well, an example of such a $B(x)$ would be $B=\nabla f$, right? Now, for such $B$, $\nabla \times B$ is zero, which is stronger than the result you want, but it seems like this might help.Ctourneur (talk) 19:19, 9 February 2009 (UTC)

If f is reasonably well-behaved (I don't know what's needed, but it's certainly enough for f to be smooth and the constant a regular value), then B must be parallel to $\nabla f$, say $\mathbf {B} =\phi \nabla f$ where $\phi$ is a scalar field, then $\nabla \times \mathbf {B} =-\nabla f\times \nabla \phi$, which is perpendicular to B, as required. Algebraist 19:28, 9 February 2009 (UTC)

Ahh brilliant - I love vector calculus but it does frustrate me so: thankyou! —Preceding unsigned comment added by Spamalert101 (talk • contribs) 02:11, 10 February 2009 (UTC)

Interest Banks pay.

The genertic function for interest is a exponential function of the form. Y = Y 0 e ^

Where Y 0 is the initial amount you put into your account, r is the interest rate, and t is the number years you have the money in your bank. Y is therefore the total amount in your account after 1 year. Find out for yourself the approximate vaule of e. What is it approximate value? Graph the equation assuming your interest rate on a corporate bond is 8%, and your initial investment is 10,000. Use a graphing program. Estimate on the graph the total value of your account after 5 years. Calculate exaclty how much money you will have after 25 years. —Preceding unsigned comment added by Freemanjr2 (talk • contribs) 16:25, 8 February 2009 (UTC)

We're not going to do your homework for you. If there is a specific bit you are stuck on, show us what you've got so far and we'll try and help. --Tango (talk) 16:30, 8 February 2009 (UTC)