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The system of equations.
Solutions have the form:
- integers asked us.For the equation:
Can specify any number :
. Then decisions will be. - any integer asked us.For the equation:
Will make a replacement that formula was compact.
- integers asked us. Then decisions can be recorded.The simple formula for the equation:
You can write this:
- integers asked us.If there is a one simple solution, it will be necessary to reduce to the corresponding
.There are other formulas. But they are bulky. Don't know whether it makes sense to write them.
I think for the equation:
It is necessary to record decisions. We will use the solutions of the Pell equation.
And then the solutions are of the form:
For the equation:
You can record a simple solution.
Where
- an odd number.This system of equations:
Solutions have the form:
- integers asked us.I decided on a different way to write the solution. These formulas do not give all solutions.
For the equation:
If we choose this number "
" so it was odd and it was not a multiple of 3.Then for and another solution:
Lay on multipliers and find
.Then
It is necessary that "
" was greater than zero. Otherwise there will be confusion.For the system of equations:
If you can the amount be represented as a sum of squares:
Then the solution can be written:
- integers asked us.I think it is better to solve this system of equations, and solutions to find the right settings.
Decisions can be recorded.
Or so:
Or so:
Then the number found as:
- integers asked us.I strive to ensure that people have used my formulas in the calculations.
To understand their use greatly simplifies the calculations.
If I write the numbers on the formula, no one will pay attention.
So. The formula I wrote the rest consider themselves.
If not satisfied then it is not necessary.
Did not understand. Decisions will be infinitely many.
For systems of equations:
Directly solved quite difficult because at first I found the solution of simple equations. And the decision recorded, so it didn't look bulky.
You can write this solution:
Number
- we will find as the solution of binary quadratic forms.The formula for the solutions of this equation there: http://www.artofproblemsolving.com/blog/101140
The number
- in the formula substituted such that the root was a rational number.The question is not as simple as it seems.
This task is equivalent to the system of equations:
It is necessary to solve the system of equations. And to find out whether there is a solution
?I think it would be an interesting case when you find out at what rate decisions will be. Very sorry. When I have time I will try to solve the system and the formula here will draw.
Did not understand.
What?
We need to write generally speaking the more General equation:
Although I formula solutions recorded, but I see it is of interest expression solutions using any one of the known solution.
If we know what any one solution:
- then you can write a formula for the solutions of this equation. - any integer asked us.For this purpose it is necessary to solve the equation:
If the next root whole
then it will be possible for this simple case record decisions: - integers asked us.For the equation:
If you choose any odd number "
"And lay multipliers.
This formula will find
Then a fairly simple solution, you can write:
It is necessary that
was greater than zero. Otherwise there will be confusion.For the equation:
You can write this simple solution:
The ratio is given for the problem. integers asked us.I became entangled in the problem and decided another equation. It is necessary to solve this equation:
Solutions will be:
integers asked us. Then the area of the quadrilateral are equal.And its perimeter is equal to:
To solve the equation:
When the number of $F$ is set for the problem. Come to the need to solve the following equation:
Left to think. What is the easiest way to solve this equation.
Review and generalize the well-known formula.
https://en.wikipedia.org/wiki/Brahmagupta%27s_formula
To begin, write the formula of the solution of the following equation:
Formulas of the solutions can be written.
The square will be equal to:
And the perimeter.
- Integers asked us.The number
determine from the equation.For not a lot of other systems of equations:
Solutions have the form:
Bad.
So much worked and solved systems of equations.
All gathered in this thread.
http://math.stackexchange.com/questions … -equations
And now. It's all removed.
For the system of equations:
You can record solutions:
- integers asked us.Although the equation is simple. But it often occurs in many cases. Required to represent the sum of the squares of the product of the multipliers. And the inverse problem. To represent the number as a sum.
Therefore, it is convenient equation to represent and solve it like this:
Then the solution can be represented as:
Or so:
- integers of any sign.It is necessary to consider not mutually simple solutions. Or Vice versa. Reduce to obtain relatively simple solutions.