How do you solve y4 - 18y2 plus 81?

Answer 1

There is nothing to solve, unless it is given as an equation.

However, the above expression may be factorised, as

y4 - 18y2 + 81 = (y2 - 9)2 = (y - 3)2 (y + 3)2 .

Should you begin, however, with the equation,

y4 - 18y2 + 81 = 0,

then, a solution for y may be found in the following manner:

y4 - 18y2 + 81 = (y2 - 9)2 = (y - 3)2 (y + 3)2 = 0;

whence, y = ±3.

But note that this solution for y holds only when the trinomial expression with which we started is valued at zero. Otherwise, it could be any finite number.

Moral: we can factorise, or otherwise process, expressions; we can solve only equations.