2.5Quadratic Equations
Section · module m51256Read §2.5 at OpenStax ↗
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Written from the section, each card quoting the span it came from. The ones you miss come back sooner.
What leans on this
Chemistry that uses this algebra — the answer to why you are learning it.
- Alongside3.3Development of Quantum Theory
The maximum electrons in shell n is 2n², and one exercise runs this backward — given a shell holding 32 electrons, find n — which is solved by isolating n² (n² = 16) and taking a square root, discarding the negative root since n must be a positive integer; recognizing that ±4 is not both physically valid is the same discipline the algebra course drills when solving quadratics by the square-root property, though small perfect squares like this one are also guessable without it.
- Illustrates13.4Equilibrium Calculations
The PCl5 ICE table is a concrete case where the quadratic formula is not busywork: x² + 0.0211x − 0.0211 = 0 has to be solved exactly, and the rejected root x = −0.156 is a root a student can explain (a negative concentration is physically impossible) in a way the algebra book's own ax²+bx+c=0 drills rarely motivate.
The PCl5 ICE table collapses to x² + 0.0211x − 0.0211 = 0, solved with the quadratic formula from Appendix B; the two roots are computed and the negative one (x = −0.156) is explicitly discarded because a concentration cannot be negative — an equilibrium calculation is unworkable without being able to run and interpret the quadratic formula.
The small-x approximation — replacing (0.15 − x) with 0.15 because x ≪ 0.15 — is an approximation criterion that only makes sense next to the exact quadratic solution it is standing in for; the section explicitly says if the resulting x fails the ≪ test, redo the calculation without the approximation, i.e. fall back to algebra:2.6.
Hydrolysis pH calculations reuse the ICE-table rational equation from equilibrium calculations — Ka = x²/(0.233 − x) for anilinium chloride — and the small-x approximation criterion, so the section is unworkable without the same quadratic/approximation toolkit already required for equilibrium ICE tables.
- Alongside14.5Polyprotic Acids
Treating the two ionization steps separately is licensed by an explicit rule of thumb (Ka1 at least 20× larger than Ka2), which is itself an approximation criterion of the same family as the small-x rule — deciding when the exact algebra (a coupled system) can be skipped in favor of two easier separate solves.
Common-ion-effect solubility calculations set up a rational equation like (0.010 + x)(x) = Ksp and rely on the same x ≪ 0.010 approximation criterion used throughout equilibrium chemistry, so the same quadratic/rational-equation toolkit is required here.