Dr Aphorism's advice

Question

My old science teacher Dr Aphorism was brilliant – not just at teaching (and, of course, science), but also at providing invaluable life advice. I always remember one day when I went to her office to talk about a coursework experiment that was not going well. She sat me down, looked at my equations, and pointed at a term involving a square root.

"I know that's the part that's wrong," I said, "but I’ve tried everything, and it just doesn’t work out."

With that, Dr Aphorism smiled and directed my attention to a poster on her wall showing a modified Periodic Table.

"I’m going to leave you here for ten minutes," she said kindly, "and I want you to focus really hard on working out the meaning behind this poster. When I come back I’m sure you’ll have a better understanding..."

TASK: What is the hidden meaning behind this poster? The (perhaps ironic) correct answer is an 8-word phrase which can be enumerated as (5, ?, 7, ?, 4, ?, 3, ?), where some word lengths have been left unknown…

^{A colour guide for colour-blind solvers is available here.}

Answer 1

First, notice that:

The numbers of colored elements are a bit off.

But if we take a look further:

The numbers are denoting other elements, which surprisingly share a common letter with the colored ones!

For example, for the RED one:

- Ni is numbered 22, denoting Ti.
- Ir is numbered 87, denoting Fr.
- Rg is numbered 118, denoting Og.
- Cf is numbered 58, denoting Ce.

But, cleverly, we don't stop here! Taking a look far further:

The numbers apparently share a common digit too!

For example, for the RED one:

- Ni is numbered 22, it should be 28.
- Ir is numbered 87, it should be 77.
- Rg is numbered 118, it should be 111.
- Cf is numbered 58, it should be 98.

So if we tabulate the results, and taking the "differences":

The last thing is to extract the final answer:

For each group, sort the digit differences, then pick the letter differences.

The final answer is:

OFTEN A PROBLEM HAS MORE THAN ONE SOLUTION.

Final remark (helped by the author):

So the wise word from Dr Aphorism to be "Remember that often a problem has more than one solution". If the narrator insists to correct the single mistake in his/her square root part, it indeed may be hard to solve or even impossible: instead, try to look at another angle!

Perhaps there are more than a single mistake here? Perhaps there is another way to solve the problem? @Johnson notices that square root may result not only positive but also negative solution (as many people forget!)

As illustrated in the poster (i.e. "there are not 1 but 2 mistakes" or "there are are not 1 but 2 solutions") surely Dr Aphorism knows how to teach her students a way of life! :)