Rewrite an equation structurally and complete the solve
Problem
Solve \(x^4~-~5x^2~+~4~=~0\) over the reals by structural substitution. Show the factorization, back-substitution, and complete solution set.
Big Picture
What this problem is really about
Structural substitution turns a higher-power equation into a familiar quadratic by naming the repeated lower power. Solving the temporary equation is only the middle of the process: every resulting value must be substituted back into the original unit. When that unit is a square, each positive value produces both a positive and a negative real branch.
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