Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A problem from 'The tangent'!!!

Well then you know a heck of a lot more than I do about it.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: A problem from 'The tangent'!!!

hi Stefy and bobbym,

I long time agothere was this problem:

Find all possible values for c if:

  a+b+c=4

and:

  a^2+b^2+c^2=8

and a,b,c are real numbers.

I have been unable to get internet access for 24 hours.  So I used some of the time profitably to work on the problem using pencil and paper.

I have only just got back home so I'm about to have dinner.

Then I will type up my musings.

Bob 

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A problem from 'The tangent'!!!

Hi Bob;

I took him through the solutions already in earlier posts but my friend does not want to do anymore of it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: A problem from 'The tangent'!!!

OK, I've seen those.

Don't you want the other solutions then?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A problem from 'The tangent'!!!

Me, heck no! But anonimnystefy will appreciate them greatly.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: A problem from 'The tangent'!!!

hi

yes,yes i would.

and bob...guess what...another miracle occurred.

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: A problem from 'The tangent'!!!

Ok bobbym.  Make sure you don't read this.

a,b,c all real

By inspection the boundary values will not work so

substitute in (1)

Note:  The two equations are 'symmetrical' in a, b and c.  By which I mean

If {a,b,c} has a solution {p,q,r} then {p,r,q}, {q,r,p}, {q,p,r}, {r,p,q} and {r,q,p} are also solutions.

I shall call this the symmetric property.

Also, because I squared the expression for 'b', equation (2) provides a  necessary but not sufficient condition for finding solutions to the original equations.

Can a = c ?

(2) becomes

This leads to two solutions, c = 2 and c = 2/3 

Substituting back we get {a,b,c}= {2, 0, 2} and {2/3, 8/3, 2/3}.

But using the 'symmetric' property this means c = 0 or c = 2 or c = 2/3 or c = 8/3.

What other solutions are there?

Suppose a + c - 2 = k then ac = k^2

The graphs below show a typical line for a + c - 2 = k and a typical curve for ac = k^2.

Both graphs have reflexive symmetry in the line y = x, so if the line crosses the curve at all then it will cross twice as shown.

This fits with the symmetric property; ie one point give (a,c) and the other (c,a).

There appears to be an infinite number of such solutions!

Bob

Last edited by Bob (2011-12-30 08:08:25)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: A problem from 'The tangent'!!!

bobbym,

You may read this:

Your avatar has disappeared completely for me. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: A problem from 'The tangent'!!!

hi bob

well that it may be that there are an infinite no. of solutions,but i just remembered what my friend did.he expressed the two relations like this:

a^2+b^2=8-c^2

and

a+b=4-c

then he drew graphs in the ab coordinate system,but i don't know what he did then.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A problem from 'The tangent'!!!

Hi Bob;

I did not read it but it is nice.

Here is what I did.

You can solve for c in terms of just a single parameter b, post#2 and post #10.

Real solutions for c depend on

and just plugging into 1 and 2 will generate them all. Since there are an infinite number they will be hard to list.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.