### Instruction: Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$.  Enter all the solutions, separated by commas.### Response: I notice that this equation has a $z^4$ term and a $z^2$ term, but no $z$ term.
This makes me think of substituting $u = z^2$ and solving for $u$ first.
If I do that, I get $u^2 - 4u + 3 = 0$.
This is a quadratic equation that I can factor as $(u-3)(u-1) = 0$.
So $u = 3$ or $u = 1$ are the possible values for $u$.
But I want to find $z$, not $u$.
So I need to undo the substitution by taking the square root of both sides.
If $u = 3$, then $z^2 = 3$, so $z = \pm \sqrt{3}$.
If $u = 1$, then $z^2 = 1$, so $z = \pm 1$.
Therefore, the solutions for $z$ are $\pm \sqrt{3}$ and $\pm 1$.