How Can I Solve x^2-xy+y^2+x+y+1=0 for My Math Project?

[AlanZa]

I need help with this Problem x^2-xy+y^2+x+y+1=0. It is part of a math project I am doing to plot a graph. The original problem was x^3+3xy+y^3=1 can factor and it plots a line at a 45 degree angle. The first part of the factorization is x+y-1=0. The only thing I can figure to do with the problem is to take the +1 and move it to the otherside of the = and make it -1 so I have a point to work with other than 0. The only number I have found that will equal -1 is to have both x and y as -1. Any help would be gladly received. Thanks for your time.

P.S. I am only in Intermediate Algerbra so please make it so I can understand.

 

[LCKurtz]

You started with your original equation:
x³+3xy+y³ - 1 = 0
and factored it to get:

(x + y - 1)(x² -xy+y²+x+y+1) = 0

and have used the first factor to get the straight line. So your remaining problem is to see if the other factor gives you anything:

x²- xy+y²+x+y+1 = 0

This is a quadratic equation in y which you can solve by the quadratic formula. So collect terms on powers of y and look at the discriminant of that quadratic equation. If the discriminant is negative you get no graph. If it is positive the quadratic formula will give you a formula for y.

 

[Mentallic]

If you're confused with what to do with the quadratic, as an alternative, try expanding:

$$\frac{1}{2}\left((x-y)^2+(x+1)^2+(y+1)^2\right)$$

and see if you can deduce anything from that.