SOLUTION: please help me graph 3x^2 + 4y^2 -6x -24y +39=0

This appears to be the equation of an ellipse:





Complete the square in the first parentheses on the left:
Coefficient of x is -2
Multiply it by 1/2, get -1
Square that: (-1)2 = +1
Add +1-1 inside the first parentheses on the end:



Complete the square in the second parentheses on the left:
Coefficient of y is -6
Multiply it by 1/2, get -3
Square that: (-3)2 = +9
Add +9-9 inside the second parentheses on the end:



Factor the first three terms in each parentheses as
the square of a binomial:



Distribute the 3 and 4 into the outer parentheses 
leaving the two squares of binomials intact:







Normally we don't get a 0 on the right, and we divide
through by what we get.  But this is a rare case.  We
cannot divide by 0.  

Both terms are non-negative, so the left side must be 0.  
It can only be 0 if x=1 and y=3.

So this is the equation of the point (1,3).  It is really
an ellipse so small it has shrunk down to a point!

Here is the graph (Yes, that's all there is to it! One point!)



Edwin

Complete squares in x and y:

 =  =  = .

So, your equation is

 = ,   or

 = .

This equation has the unique solution: x =1 and y=3.
There is no other real solutions.

So, the equation describes only one point (1,3).

The plot consists of only one point (1,3).