Sketch the graph of x^2+y^2-6x-4y=23

Firstly, to try and sketch this graph just going on the equation as it is above is a huge task for any maths student. The best way to approach this is to neaten it up with some algebra:

-1: x^2+y^2-6x-4y can be grouped into 'like terms': x^2-6x-+y^2-4y=23

-2: recall that we can factorise this and make up for any inaccuracies in our factorising: for example, x^2-6x=(x-3)^2-9 (the minus 9 compensates for the unwanted constant term coming out of this equation). Doing this with both x and y yeilds: (x-3)^2-9+(y-2)^2-4=23

-3:This looks a lot nicer! Taking the constant terms all to one side we get: (x-3)^2+(y-2)^2=36

-4: Finally, we have to consider what this equation looks a little bit like. It's almost in the form x^2+y^2=r^2, which is the equation of a circle, centred at the origin (0,0) with radius r. Fortunately, the RHS of the equation can be compared nicely to see that the radius of the circle is the positive (since radii are never negative) square root of 36, or 6. Looking at the left side however, we see that our x and y are subtracted from before being squared. Not a problem. All this means is that any x value we put in ie. x=1 will be shifted by -3, which is a shift to the right on the graph. And a shift up by two units on the graph in the case of y. The resulting graph is a circle, centred at (3,2), with radius 6.