Question

solve \(3x^2+3xy-5y^2=55\)

Answer 1

x,y are real numbers

Answer 2

This a curve. What do you mean by solve it.

Answer 3

http://www.wolframalpha.com/input/?i=plot3d+3x^2%2B3xy-5y^2%3D55

Answer 4

Print it out, put the paper in water or some other kind of "solvent" to dissolve it. Problem "solved"

Answer 5

It is a hyperbola

Answer 6

i believe we are loocing for pairs \(x,y\) i have found such pairs for integers but i am concerned with none integers,real numbers

Answer 7

^^^looking

Answer 8

irs should satisfy the hyperbola

Answer 9

There are infinite numbers of (x,y) that satisfy your equation. They are lie on the hyperbola drwan by wolframalpha
http://www.wolframalpha.com/input/?i=plot3d+3x^2%2B3xy-5y^2%3D55

Answer 10

@Jonask You should review multivariable functions.

Answer 11

for exam purposes i want to get that only with a calculator,how do i proceed

Answer 12

http://www.wolframalpha.com/input/?i=3x%5E2%2B3xy-5y%5E2%3D55%2Cx%2Cy+in+integer+

Answer 13

ohh this is a number theory class,not calculus

Answer 14

You have x and y, so there'll be many solutions for each.

Answer 15

@primeralph

Answer 16

Yeah, either way you need to tell us what you're solving.

Answer 17

The points
(4, 1)
(-5, 1)
satisfy the equation

Answer 18

well basically all solutions for example>>>\(\dfrac{27+\sqrt{69}}{2}\)

Answer 19

i have found integer solutions here http://math.stackexchange.com/questions/739752/integer-solutions-of-the-equation-of-the-form-ax2bxycy2-m#

Answer 20

concerned with none integer solutions,ie real numbers

Answer 21

NONE INTEGER SOLUTIONS

Answer 22

more generally the book gives solutions

\(\pm(\dfrac{17\pm\sqrt{29}}{2})(2+\dfrac{1\pm\sqrt{29}}{2})^{2n} \cup (\dfrac{41\pm7\sqrt{29}}{2})(2+\dfrac{1\pm\sqrt{29}}{2})^{2n}\)

\[n\]is integer

Answer 23

Here is a list of all integer solutions betwee -100 and 100
{{-100, 53}, {-53, 28}, {-47, -53}, {-25, -28}, {-25,
13}, {-12, -13}, {-9, 4}, {-5, -4}, {-5, 1}, {-4, -1}, {4,
1}, {5, -1}, {5, 4}, {9, -4}, {12, 13}, {25, -13}, {25, 28}, {47,
53}, {53, -28}, {100, -53}}

Answer 24

Here is a list of all integer solutions between -1000 and 1000
{{-620, -701}, {-620, 329}, {-291, -329}, {-213,
113}, {-100, -113}, {-100, 53}, {-53,
28}, {-47, -53}, {-25, -28}, {-25, 13}, {-12, -13}, {-9,
4}, {-5, -4}, {-5, 1}, {-4, -1}, {4, 1}, {5, -1}, {5,
4}, {9, -4}, {12, 13}, {25, -13}, {25, 28}, {47,
53}, {53, -28}, {100, -53}, {100, 113}, {213, -113}, {291,
329}, {620, -329}, {620, 701}}

Answer 25

@eliassaab how can we generalise the integer solutions,and how did you find other integer solutions?

Answer 26

I wrote a Mathematica program to generate the list. Change p and the program below and execute
Clear[f, x, y]
p = 1000;
f[x_, y_] = 3 x^2 + 3 x y - 5 y^2 - 55;
Q = {};
For[ m = -p, m <= p, m++,
For[ n = -p, n <= p, n++,
If[f[m, n] == 0, Q = Append[Q, {m, n}]]]]
Q

Answer 27

ok,i understand integer solutions,sir do you know how we can find---real number solutions,i mean non integers

Answer 28

Take a any real number, replace y by a. You will have a quadratic in x, solve it in terms of a and you are done if your discriminant is positive or zero

Answer 29

There are infinite number of them.

Answer 30

That is what you get
\[
\left\{\left\{x\to \frac{1}{6} \left(-\sqrt{3} \sqrt{23 a^2+220}-3
a\right)\right\},\left\{x\to \frac{1}{6} \left(\sqrt{3} \sqrt{23
a^2+220}-3 a\right)\right\}\right\}
\]

Answer 31

yes \[\Delta=69\] but what steps are necessary to do this
\(O_{69}=[1,\omega_{69}]=[1,\frac{1+\sqrt{69}}{2}]\)

Answer 32

You can see that the discriminant of the quadratic is always positive

Answer 33

so you take a as a constant,then treat it as a normal quadratic of the form \[\alpha x^2+\beta x+c=0\]

Answer 34

Yes

Answer 35

i would be interested i finding \(a\) as well

Answer 36

This a list of some non integer solutions
\[

\left(
\begin{array}{cc}
\frac{1}{6} \left(\frac{3}{13}-\frac{3 \sqrt{12401}}{13}\right) &
-\frac{1}{13} \\
\frac{1}{6} \left(-\frac{9}{130}-\frac{\sqrt{11154621}}{130}\right) &
\frac{3}{130} \\
\frac{1}{6} \left(-\frac{24}{65}-\frac{6 \sqrt{77581}}{65}\right) &
\frac{8}{65} \\
\frac{1}{6} \left(-\frac{87}{130}-\frac{3 \sqrt{1245781}}{130}\right) &
\frac{29}{130} \\
\frac{1}{6} \left(-\frac{63}{65}-\frac{\sqrt{2818929}}{65}\right) &
\frac{21}{65} \\
\frac{1}{6} \left(-\frac{33}{26}-\frac{3 \sqrt{50501}}{26}\right) &
\frac{11}{26} \\
\frac{1}{6} \left(-\frac{102}{65}-\frac{42 \sqrt{1626}}{65}\right) &
\frac{34}{65} \\
\frac{1}{6} \left(-\frac{243}{130}-\frac{\sqrt{11606709}}{130}\right) &
\frac{81}{130} \\
\frac{1}{6} \left(-\frac{141}{65}-\frac{3 \sqrt{326769}}{65}\right) &
\frac{47}{65} \\
\frac{1}{6} \left(-\frac{321}{130}-\frac{3 \sqrt{1327109}}{130}\right) &
\frac{107}{130} \\
\frac{1}{6} \left(-\frac{36}{13}-\frac{2 \sqrt{30369}}{13}\right) &
\frac{12}{13} \\
\end{array}
\right)

\]

Answer 37

They are all of the form
\[
\left\{\frac{1}{6} \left(-\sqrt{3} \sqrt{23 a^2+220}-3 a\right),a\right\}
\]
Vary a and you get the other cooerdinates

Answer 38

If you want of find a as well, put x = b and solve the quadratic in terms of b

Answer 39

\[
\left\{\left\{y\to \frac{1}{10} \left(3 b-\sqrt{69
b^2-1100}\right)\right\},\left\{y\to \frac{1}{10} \left(\sqrt{69
b^2-1100}+3 b\right)\right\}\right\}

\]
Here you have to chose b to have the discriminant positive

Answer 40

hmmm ok, this is what i wanted ,so logically this would mean we are looking for values where the level curve's achieve a value of 55 for \(f(x,y)=3x^2+3xy-5y^2\),thank you @eliassaab

Answer 41

by the way you said \(a,b\) are real numbers,it is therefore hard to generate such numbers,that will gve a positive dirsciminant

Answer 42

if it was integers then it would be easy to generate such integers

Answer 43

psh, ur all mathletes