Show me where y = x^2 +1 crosses the X-Axis visually

Off topicOther → Show me where y = x^2 +1 crosses the X-Axis visually

CHALLENGE

I know how this can be visualized, but I want to see how other people try to.

I mean, there’s several ways to do this. The three that come to mind would be with quadratic formula, completing the square and factorising.

Quadratic formula

x=[-b±√(b²-4ac)]/2a x=[0±√(-4)]/2 x=±2i/2 x=±i

Completing the square

x²+1=0 x²=-1 x=±i (much easier than normal since coefficient of x is 0)

Factorising

Not much to say here tbh (x+i)(x-i)=0 => x=±i

I know there’s probably other ways to do this, but this is how I usually solve quadratics

Sorry, I meant visualize (updated title)

Here is a picture of it crossing the x-axis on the x-y plane:

Is that good enough?

Consider the 3d graph: z = (x+yi)^2 + 1 which is essentially the same as y = x^2 + 1, just with the x replaced with the complex number x+yi, so the x coordinate is the real axis and the y is the imaginary axis.

when y = 0, z = x^2 + 1 so the graph is a parabola moved up by 1 on the real axis.

when x = 0, z = -y^2 + 1 so the graph is an upside-down parabola moved up by 1 on the imaginary axis. This means the graph cuts the plane z=0 like an upside-down parabola would cut the x axis in cartesian coordinates, except in the imaginary direction instead.

This is the graph of z = (x+yi)^2 + 1:

alt text If you only look at the plane through x=0 you can clearly see an upside-down parabola which cuts at i and -i.

Yes, but that’s a lot more than 2 points, your missing y’s imaginary.

It doesn’t cross the X axis. It only has imaginary roots. Anyways, you can think of it as if there’s an imaginary parabola which is upside down, with the equation y = -x^2 + 1

The points where that parabola crosses the X axis are the imaginary roots. (-i, i)

Yes viper text = visually good job you can read. The entire point of this thread was to see how people choose to visualize it.

Also it does cross, the extension of imaginary numbers (badly named) is an extension to the number line, like any other extension in history.

Edit: A better explanation is above, but just because it doesn’t crosses x’s real axis, that doesn’t mean it’s not crossing the x-axis. That’s like you didn’t pass a house, just because you were flying above it.