We know 1 <= x <= 3
Squaring both sides gives you 2(1 + ¡î(x-1)(3-x))=(x©÷ - 4x + 6)©÷
We know x©÷ - 4x + 6 ranges from 2 to +¡Ä, so that squared ranges from 4 to +¡Ä
We also know that ¡î(x-1)(3-x) ranges from 0 to 1 (maxing out at x=2), so 2(1 + ¡î(x-1)(3-x)) ranges from 2 to 4 (maxing out at x=2)
If the left half ranges from 2 to 4 and the right half from 4 to +¡Ä, the only possible intersection is when both halves equal 4, (making both halves of the original equation equal 2).
Set one side of the original equation to 2 and you get the only solution, x=2.
Posted by Peter
on 2006-12-11 20:56:52