Does x^2 + y^2 = -4 have a graph? If so, what is it? If not, why not?

Answer 1

No it doesn't. (Not over the real numbers anyway....)

x² and y² must be positive or zero.

The sum of 2 positive numbers is positive, and 0 + 0 = 0. You can't get -4.

Equations that look like "x² + y² = r²" are the graphs of circles with a radius r. That equation is for a circle with whose radius would be 2i (two times the square root of negative one, an imaginary number).

Answer 2

Since x^2 and y^2 will be positive for any point (x, y), then x^2 + y^2 is never equal to -4. So no points (x, y) satisfy the equation.

Yes, this equation does have a graph. Its graph is the empty graph (the graph with no points on it).

Answer 3

...

x^2 + y^2 = -4 does not have a graph because the equation is impossible. When you square any number, the result is a positive number. When two positive numbers are added, the final result should be a positive, not negative.

...

Answer 4

I agree with him. There is a plane where complex numbers can be shown. so we can even solve x^2+1=0 equation