IMHO, yes, fire the second pool company. The first company deserves to be reported to the BBB.
Backwashing does NOTHING to kill algae.
Yes, it's easy to maintain your pool yourself so you NEVER have this problem again, save thousands of $$$ over the years. But you'll have to clean up the mess first. And you will have to learn the basics.
As hard as it is for me to say this your pool store is bang-on correct about the 2nd company AND their advice to shock the pool.
When they tested your water, did they give you a print-out of the results? If so, please list them for us. We are particularly looking for the following:
FC
CC
TC
pH
TA or Alk
CYA
CH
Also tell us about your pool. How many gallons is it? Is it vinyl, fiberglass, or concrete/shotcrete/gunite/etc.? What brand and model pump and filter do you have? What size are they? What type of filter: Sand, DE or cartridge?
Also, go to our sister site, PoolSolutions.com and start reading--it should take about an hour's reading but pretty much everything we do is there.
In short what you'll need to do is boost chlorine levels high enough to kill the algae and then maintain your water correctly. You'll need to learn to test your water, too. It's easy, but a critical step to "Pool Independence"
Pool maintenance comes down to a very few things:
Chlorine levels
pH (remember that from HS chemistry)
Consistency.
Pool stores have a zillion chemicals they try to sell. Some are available by other names from the supermarker. REALLY! Others do no good, but they push them anyway (Like calcium in a vinyl pool. Useless). Others are scare tactics (Phosphate levels are too high! BUZZZ--rarely a problem). Others are just bad, like spraying air spray on dog poop rather than cleaning it up. Others are worse and make YOUR problem, algae much, MUCH harder to clean up.
Most of us, use in our pools:
Bleach or Liquid Chlorine, same thing as household bleach only more concentrated.
Borax--yeah, 20 Mule Team Borax
Baking Soda--Arm&Hammer
Washing Soda--Also Arm&Hammer
Muriatic Acid
From pool stores we use:
CYA, called Stabilizer or Conditioner
Polyquat, the ONLY algaecide you should EVER use.
Dry Acid, in lieu of Muriatic Acid.
We call this the B-B-B method for Bleach, Borax and Baking Soda, and it really, really works. ALL of the grocery store chemicals are sold by the pool stores at higher prices with fancy names but are EXACTLY THE SAME!
Welcome! Get us the info and we'll tell you where to start and what to do.
Carl
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