Don't dictate what is going to happen.
The quickest way to turn your child against any idea, no matter how rationale and well-intentioned, is to dictate. Telling her that she will be going to private school will probably not get the result you want. Think about her feelins. She has friends. They may not be the friends you might prefer her to have, but they are friends nonetheless. Sending her to private school changes the balance of her comfortable little world. Remember that teens are particularly sensitive to change. Take her out of her comfort zone and she will be most unhappy.
She must buy into the idea.
The minute she decides that going to private school is a great idea, you will be set. I realize that I sound like a manipulating adult, but that's pretty much what it takes to achieve your goal. Neither can you present the idea to her like you present a proposal to a client. You must be subtle to achieve your goal.
There are several approaches you can take. If you or your spouse has had a promotion and the new job requires more travel and less quality time at home, lay out your concerns. Do it while driving around doing your Saturday errands. Stop for a coffee and sit down to enjoy a precious few moments with this rapidly-growing child of yours. If you listen to what she is talking about, you might hear an opening for discussion of school. Seize that opportunity to begin the discussion. On her terms.
Now, if your child is struggling academically, then attending private school usually is an easier thing to sell. The discussion might begin with the fact that failing grades in math and science will limit her choices when it comes to getting into college. It shouldn't take much to convince her that the one-on-one help which she will receive in the small classes at private school will be a good thing.
Is your child gifted? That is also another clear-cut reason for suggesting that she go to private school. Many times gifted children - also known as nerds - can have a tough time fitting in to a socially stratified public high school where you are a jock or a geek or a loser. Those kinds of labels just won't fly in a private school. Tolerance and appreciation of learning differences and styles is much more prevalant in private school community simply because the community doesn't have to deal with student rights and due process. When you sign the contract with a private school, those rights you might have had in a public school are replaced by something known as contract rights. The contract details very clearly the consequences for not living up to the terms of the contract. Private schools have zero-tolerance for students who break the rules. In other words it's cool to be smart in private school. It's just part of what private schools are.
Does your child have a learning difference? This once again is a sound reason why private school would be a good choice for him. For example, If he has struggled with dyslexia, he will feel much more comfortable, indeed 'normal', when he is being taught by experienced, highly credentialled teachers who understand what he is coping with and know how to
remediate his syndrome.
Does your child excel in the arts or sports? It shouldn't be too difficult to convince him to go to private school. Make it his idea by taking him to a concert or game at a private school. It could just be the eye-opening experience which he needs to make it his idea to go to private school.
One final strategy worth mentioning. The Bible wisely states that 'There is no prophet left without honour except in his own country and among his own family'. With that in mind why not have a trusted family friend do the heavy lifting for you? After all, she is probably more credible than you are as far as your child is concerned. Then when your child announces that Aunt Sheila thinks going away to private school is a great idea, all you have to do is handle the details. The idea has been sold. Your child has been convinced to go to private school.