My DDs had some pretty significant gaps in the knowledge between K12 and SOS, so we started using Math U See with good results. I like it.
Stepping on my soapbox...she is not your "friends'" child. She is YOUR child. It's none of their business how you choose to school her. You can actually teach her MORE than the school can. My DDs have learned about Michelangelo, Botswana, photography, and more, none of which they would have EVER been able to learn at all, much less in elementary school. The "socialization" issue is a weak response of frightened people who have been socially engineered through the public school model to believe that there is only one right way to educate children.
There is a plethora of activities for homeschooled children providing true socialization. When was the last time you had a job that was ONLY 44-45 year olds? When was the last time you were in the breakroom at work only to be pushed down and your lunch stolen? When was the last time you were in the restroom and had someone offer you marijuana? Precisely NOT the kind of "socialization" I'm interested in, thankyouverymuch. True socialization comes from interacting with people of all ages in positive and healthy ways, even if there is conflict.
In our experience, my 2 older DDs started out in PS (I was a high school English teacher, after all...) and it was VERY difficult to pull them out and re-educate them because of the social programming that had already taken place. We struggled with our decision, even though another child had raged and threatened to kill my daughter and they were only in the 2nd grade. He received no discipline and was allowed to continue to harass her, so we pulled them both out of the school. My younger 2 have never been in a "school" and have done very well. We are a hybrid of "unschooling" and "homeschooling" in that we use some curriculum, but a LOT of what my DDs learn is interest-based and self-driven. We had a lizard in our house this week, so it was straight away to Google to see what we could learn. We didn't know what kind of lizard it was at first, but DDs did some quick searching and discovered that is was a Mediterranean Gecko. Lo and behold, it's not native to Florida and is the only nocturnal lizard we have here. We've recently worked on a lapbook about animal classification, so they were able to add the information about our new "friend" to their books.
My educational philosophy is, "Wake up. Start learning." My second philosophy is "I've SEEN the village. I don't want their help, thank you." My third? "Mind your own beeswax"

You have to make the choice that is best for YOU, YOUR child, and YOUR family. It may very well be a public, private, or charter school, but understand that there is a degree of de-schooling that has to happen if they start out that way. I always say it's easier to start them at home and put them in school later than the other way around.