The normal quadratic you solve in school is [math]y=ax^2+bx+c[/math][br][br]Let's simplify and call it [math]y=x^2+c[/math] so it is centred around the [math]y[/math] axis and to make the visualisation easier. Now change [math]y[/math] to [math]z[/math] and replace the ordinary number [math]x[/math] with the complex number [math]x_c[/math]. Also replace [math]c[/math] with [math]p[/math] to give us [math]z=x_{c^2}+p[/math].[br][br]We know the complex number [math]x_c=x+iy[/math] so substitute:[br][math]z=\left(x+iy\right)\left(x+iy\right)+p[/math][br][math]z=x^2+ixy+ixy+i^2y^2+p[/math][br][math]z=x^2+2ixy-y^2+p[/math][br]Now ignoring the imaginary part we get[br][math]z=x^2-y^2+p[/math][br][br]This is plotted on the axis below along with two planes, [math]x=0[/math] and [math]y=0[/math]. Where these planes intercept the surface, we can see the standard 2D quadratic plots that you might be used to from school. As you move slider [math]p[/math] you can see that at no point is there no solution to the initial quadratic.[br][br](Thanks to [url=http://www.bikinfo.com/HTML/quadratic.html]bikinfo[/url] for the initial idea to create this visualisation)