Answer
The x-intercepts are (-1,0) and (0,0)
The y-intercepts are (0,-2) and (0,0)
The equation has no symmetry with respect to the x-axis, y-axis, or origin.
Work Step by Step
To find the x-intercept(s), we set y to 0 and solve for x:
$x^2+x+0^2+2(0)=0$
$x^2+x=0$
$x(x+1)=0$
$x_1=0$
$x_2+1=0\rightarrow x_2=-1$
To find the y-intercept(s), we set x to y and solve for y:
$0^2+0+y^2+2y=0$
$y^2+2y=0$
$y(y+2)=0$
$y_1=0$
$y_2+2=0\rightarrow y_2=-2$
To test for symmetry with respect to the x-axis, we substitute y for -y and check if it equals the original equation:
$x^2+x+(-y)^2+2(-y)=0$
$x^2+x+y^2-2y=0$ nope
To test for symmetry with respect to the y-axis, we substitute x for -x and check if it equals the original equation:
$(-x)^2+(-x)+y^2+2y=0$
$x^2-x+y^2+2y=0$ nope
To test for symmetry with respect to the origin, we substitute x for -x, substitute y for -y and check if it equals the original equation:
$(-x)^2+(-x)+(-y)^2+2(-y)=0$
$x^2-x+y^2-2y=0$ nope
