How can I imagine double/repeated root of a quadratic equation?

Question

A quadratic equation such as $(x-2)^2=0$ has a repeated root $(2,2)$. A lot of things in math (equations, matrixes and so) can be nicely drawn on a $2D, 3D$ etc plane (with $x$-$y$ axis).

I mean, I though there is a relation between algebraic equations and geometry, but in this equation, $(x-2)^2$ is a parabola with clearly only one point that lies on an $x$ axis? How could I understand, that there are two roots then? Also with the imaginary roots I have found ways how to draw it, but it was just something like - how to find the roots with geometry, not something as logical and simple as when a parabola graph penetrates $x$ axis.

Is it because the real result is not exactly a parabola that we can draw on a $x$-$y$ plane or why?

Just for info - I know how to solve it and all the rules, I am really just asking on how to understand it geometrically.

Thanks a lot

Answer 1

Try to interpret (x-2)^2 = 0 as the intersection of y = (x-2)^2 and y = 0. You will find the latter is tangent to the parabola.

You can further imagine that one half of the tangent is running from – infinity towards (2, 0) and the other half is running from the + infinity towards the (2, 0).

Answer 2

Easiest way to understand it imo, is to consider the quadratic function

$f(x)=(x-2)^2+c$

Edit: I think it's important to point out here that changing the value of $c$ translates the graph of $f$ vertically.

For small values of $c<0$. You will notice that there are two roots, but that as $c$ gets closer to $0$ those roots are getting closer not only to each other, but also to the vertex of the parabola. When $c=0$ the roots now coincide, and you have what is known as a double root or repeated root. (This concept can be further generalized for higher degree polynomials and is known as multiplicity of a root; it conveys certain information about relative behavior of the graph around the given root.)

If you continue to considering $c>0$, you no longer have real roots, but there are complex numbers which can still satisfy the equation.