Why Was Snape's Potions intructions superior to the Book?

In Harry Potter and the Half-Blood Prince, Harry acquires an old textbook once belonged to the mysterious Half-Blood Prince. After first being annoyed with the scribbles of the book, Harry found them to be instructions more useful than those in the textbook. But Why?

Why are Snape's instructions correct while the official instructions appear to be wrong?

Here is a link to a Quora Page discussing this issue. Most of the answers appear to agree with a single explanation:
https://www.quora.com/Why-were-the-textbook-potions-wrong-and-how-did-Snape-know-the-correct-formulae

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