Every Quadratic Surrenders

Unit 4 ends with a guarantee — every quadratic equation surrenders its solutions — and the unit's real subject is custody: no root lost, none divided away, none fabricated and left unaudited.

The methods form a hierarchy of reach. Factoring is fastest where the pair search lands: the zero-product property splits (x − 3)(x + 2) = 0 into its two answers on sight. The square root property handles the bare squares, and its ± is not decoration — x² = 9 owns 3 and −3 both. Completing the square works on everything, manufacturing the perfect-square pattern it needs; and the quadratic formula is completing the square run once for all time, which is why it never fails. Before it even finishes, the discriminant calls the ending: 49 promises two real solutions, zero promises one repeated, and −3 promises a conjugate pair. That last promise needed new numbers. The imaginary unit — one definition, i² = −1 — completes the story the reals left hanging: √(−4) becomes 2i, x² + 4 = 0 finds its pair, and the sum of squares that refused to factor splits at last as (a + bi)(a − bi). Nothing old broke; the property list still holds, and the arithmetic of i is FOIL plus one substitution. Against the guarantee stand the unit's three custody crimes. The lost root drops the minus sign that x² = 9 is owed. The divided-away solution erases x = 0 by dividing where factoring was due. The unchecked squaring reports −1 as a root of √(2x + 3) = x, when √1 was never −1. Each dies the same death — a check in the original — and that is the unit's law: candidates are cheap, and the original equation is the only judge.