Why does orbital hybridization take place?

I will try to explain hybridization from its simplest concept.

Keep in mind that hybridization occurs to satisfy the molecular geometry in order to achieve the lowest state of energy of the molecule.

Let us consider the simplest example which is methane: #CH_4#.
Someone, might imagine that for four hydrogen atoms to be around the carbon atom the Lewis structure would be this:

The way that this molecule is drawn, the bond angles are #90^@# and the geometry of the molecule is square planar. We could justify this by saying that this is normal since carbon will be using its #s# orbital and #p_x, p_y and p_z# orbitals where the angles in between are #90^@# because they are oriented through the #x, y and z# axises.

However, the experimental results have shown that the molecular geometry around a carbon atom is tetrahedral and that the single bonds angles are #109.5^@#

which cannot happen if the carbon atom uses its #p_x, p_y and p_z# orbitals, therefore, the need of forming NEW ORBITALS which will be the result of mixing the #s# orbital and #p_x, p_y and p_z# orbitals all together to get new four orbitals #sp^3#. In a similar way, we can discuss the #sp^2#, #sp#, etc.

Therefore, we can use the following:
Tetrahedral: #sp^3#
Trigonal planar: #sp^2#
Linear: #sp#