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Thread Subject:
how to solve an implicit equation

Subject: how to solve an implicit equation

From: Peter Schreiber

Date: 27 Sep, 2011 20:14:11

Message: 1 of 4

Hello,
How would I go about solving an implicit equation like..


sqrt(a*x^2+b*x+c)+sqrt(d*x^2+e*x+f)=g

a,b,c,d,e,f,g are real constants and I would like to find x (which is a real quantity too) that solves the equation.

Any help is highly appreciated. I don't think that there is an analytical solution to this problem, but I would prefer a numeric solution anyway.

Cheers,
Peter

Subject: how to solve an implicit equation

From: Bruno Luong

Date: 27 Sep, 2011 21:33:29

Message: 2 of 4

"Peter Schreiber" <schreiber.peter15@gmail.com> wrote in message <j5taqi$bsr$1@newscl01ah.mathworks.com>...
> Hello,
> How would I go about solving an implicit equation like..
>
>
> sqrt(a*x^2+b*x+c)+sqrt(d*x^2+e*x+f)=g
>
> a,b,c,d,e,f,g are real constants and I would like to find x (which is a real quantity too) that solves the equation.
>

Let
P(x) := a*x^2+b*x+c
Q(x) := d*x^2+e*x+f.

The above equation is:

sqrt(P(x))+sqrt(Q(x))=g

=> P(x) + Q(x) + sqrt(P(x)*Q(x)) = g^2
=> P(x)*Q(x) = [g^2 - P(x) + Q(x)]^2
=> [g^2 - P(x) + Q(x)]^2 - P(x)*Q(x) = 0

The latest equation is a quartic polynomial equation that can be solved numerically with ROOTS command. A close form solutions also exist.

Bruno

Subject: how to solve an implicit equation

From: TideMan

Date: 27 Sep, 2011 22:38:54

Message: 3 of 4

On Sep 28, 10:33 am, "Bruno Luong" <b.lu...@fogale.findmycountry>
wrote:
> "Peter Schreiber" <schreiber.pete...@gmail.com> wrote in message <j5taqi$bs...@newscl01ah.mathworks.com>...
> > Hello,
> > How would I go about solving an implicit equation like..
>
> > sqrt(a*x^2+b*x+c)+sqrt(d*x^2+e*x+f)=g
>
> > a,b,c,d,e,f,g are real constants and I would like to find x (which is a real quantity too) that solves the equation.
>
> Let
> P(x) := a*x^2+b*x+c
> Q(x) := d*x^2+e*x+f.
>
> The above equation is:
>
> sqrt(P(x))+sqrt(Q(x))=g
>
> => P(x) + Q(x) + sqrt(P(x)*Q(x)) = g^2
> => P(x)*Q(x) = [g^2 - P(x) + Q(x)]^2
> => [g^2 - P(x) + Q(x)]^2 - P(x)*Q(x)  = 0
>
> The latest equation is a quartic polynomial equation that can be solved numerically with ROOTS command. A close form solutions also exist.
>
> Bruno

Uh, oh, Bruno, you've missed a 2 factor in the cross product.
(a+b)^2=a^2 + 2ab +b^2

Subject: how to solve an implicit equation

From: Bruno Luong

Date: 28 Sep, 2011 06:05:27

Message: 4 of 4

TideMan <mulgor@gmail.com> wrote in message <93340337-6e76-4f9b-b33e-37378c3338ce@i33g2000yqm.googlegroups.com>...

>
> Uh, oh, Bruno, you've missed a 2 factor in the cross product.
> (a+b)^2=a^2 + 2ab +b^2

Oops thanks TideMan

Bruno

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