What information does a complete ionic equation give that the balanced equation doesn't show?

Answer: It shows how the compounds actually exist in solution.

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A complete ionic equation shows ions formed from each of the species. It also show the charge on each of the ions. A chemical equation shows all reactants as compounds or elements. The ions formed and their charges can only be inferred. Explanation

For example, NaCl reacts with AgNO₃ to produce NaNO₃ and AgCl (a precipitate). The chemical equation for this reaction will be:

NaCl + AgNO₃ → NaNO₃ + AgCl (s)

The complete ionic equation for the same reaction will be:

Na⁺ + Cl⁻ + Ag⁺ + NO₃⁻ → Na⁺ + NO₃⁻ + AgCl (s)

Excepting for the AgCl solid, all other species in the ionic equation are written as ions.

NaCl exists as Na⁺ ions and Cl⁻ ions. AgNO₃ exists as Ag⁺ ions and NO₃⁻ ions. NaNO₃ exists as Na⁺ ions and NO₃⁻ ions. AgCl does not dissolve in water. Rather, its ions stay intact within the ionic lattice.

Their subscripts show their charges. There's no subscript in the chemical equation. It takes some reasoning to tell the ions formed from each of the species.