# Equation "solved through" or "solved by"?

Which one is better?

• "The equation x² = 4 is solved through 2 and -2."
• "The equation x² = 4 is solved by 2 and -2."

Which other suggestions do you have?

Just by googling, I could not resolve this problem...

• This suggests solved by books.google.com/ngrams/… - if you run the function over a range of numbers it would be The equation is run from -2 through 2 May 9, 2014 at 9:29
• The equation x² = 4 has the solutions 2 and -2. May 9, 2014 at 9:47
• "The equation x² = 4 is satisfied by x = -2 and x = 2." // "The equation x² = 4 has the roots -2 and 2." May 9, 2014 at 10:04
• Thanks for your very helpful comments. I have increased the complexity of my question a bit in a comment given to bsink's answer. May 9, 2014 at 10:25

## 1 Answer

Of the two, I personally think "solved by" sounds better.

The comments on your question provide some excellent alternatives as well. "has the solutions/roots" and "is satisfied by" are great, though I would prefer the former if you're supplying the full list of possible solutions as the latter could be ambiguous. E.g. "x²=4 is satisfied by x=2" is completely accurate, although it doesn't provide all possible roots to the quadratic.

• Thanks a lot for your answer! Could you help me with something like "The equation system a + b = 2, 2b = 2 is successively solved in a and b by b = 1, a = 2 - b = 2 - 1 = 1" (seems to be a little bit more constructed, but I have this construction in a more complex context)? Should it be "by a applying", "through" or something else? May 9, 2014 at 10:22