Deep Dive: Discriminant With a Parameter

*Provenance: compiled 2026-09-29. Read in full: research/veridian/WMA11-verified-facts.md (grepped for "discriminant" and "real root", then read at §1's spec table, §4.1's own discriminant-with-parameter entry, the discriminant-with-a-parameter.ts mark-scheme-bullet coverage audit inside "Lesson-audit log" §, §6 "Topics with enough verified material for a genuine standalone lesson," and §7 "Flagged issues"); the live lesson file content/veridian/curriculum/wma11/discriminant-with-a-parameter.ts (read in full — header, all prequestions, both teach blocks, the mechanism block, diagram, worked-chain, chain-drill, marked- solution, all seven trap-taxonomy items, the beyond-spec block, all five MCQs, reference card and flag); signal-topic-brief-discriminant-with-a-parameter.md (read in full); signal-frequency-and-boundaries.md (grepped, one line used); codex-formula-and-method-bank.md §1.1.3 and §2.1.2 (read in full); codex-complete-worked-paper.md's own Question 4 (read); forge-error-pattern-tracker.md §1's EP1a–EP1e codes, forge-battle-card.md, forge-exam-technique-guide.md and forge-vocabulary-precision-list.md (each grepped for "discriminant" and cross-checked); content/veridian/question-bank/wma11.ts (checked for overlap — no discriminant-with-parameter item is seeded there yet). No new primary-source PDF was fetched for this document; every real-exam fact below is drawn from citations already verified in the sources just listed, and every arithmetic step shown (discriminant expansions, factorisations, the general parameter- quadratic formula in §2 and its check against both real anchors) was independently recomputed by hand before being written in here, not copied blind.

#What this document is, and isn't

A single-topic reference for spec point 1.4's parameter variant only. It does not repeat the lesson's full worked chain (3px^2+12px+4=0), its chain-drill (x^2+tx+(t+3)=0), its marked-solution (x+4/x=c), its diagram, or its five MCQs stage-by-stage — those already exist, are pedagogically paced, and use numbers chosen specifically to differ from the two real anchors (per the lesson's own header). What follows instead is: the general theory stated as compact algebraic reference, including one genuinely new piece of it (§2's formula for how the discriminant behaves as a function of the parameter, and what determines its degree); the real distinct sub-cases this exact spec point has actually been tested from, worked through on the real anchor numbers themselves (which the lesson deliberately avoids using); two further sub-cases the general theory predicts but no real anchor has yet confirmed, worked through on short fresh micro-examples so the gap is demonstrated rather than merely asserted; and a single consolidated synthesis of every real WMA11 exam appearance of this topic this course's research has actually found — with the evidentiary gaps stated as plainly as the findings.

Two labels are used consistently below:

• [real] — a verified fact from a genuine Pearson question paper, mark scheme or examiner report, with series and question number given.
• [VERIDIAN, illustrative] — a fresh invented micro-example built to demonstrate a mechanism, with no claim to resemble any specific real Pearson question — the same disclosure convention the lesson file itself uses throughout.

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#1. What spec 1.4 actually asks for

The specification states this point in one short clause (spec P1.3, item 1.4): candidates are expected to "know and use $b^2-4ac>0$, $=0$, and $<0$." Two things follow directly from that wording. First, the spec names no restriction on what $a$, $b$ or $c$ are allowed to be — a parameter sitting inside any one of them is not a separate spec point, it is the same spec point applied to a coefficient that happens to be a letter instead of a number. Second, the spec's own three conditions are given as strict/zero/strict-the-other-way — $>0$, $=0$, $<0$ — which is exactly why "no real roots" and "two distinct real roots" are both strict conditions on the boundary, while a real question asking for "at least one root" is really asking for the union of two of the spec's three named cases ($>0$ or $=0$), not a fourth case the spec introduces. This topic sits downstream of spec 1.7–1.9 (quadratic inequalities) for exactly the reason the lesson's own mechanism block states: once a parameter is involved, applying the spec's three conditions means solving an inequality in the parameter, and that solve uses the critical-value-then-sign-test method spec 1.9 already covers, one level removed from $x$.

#2. The general mechanism, stated as compact reference

The basic three cases, exactly as spec 1.4 states them: $\Delta=b^2-4ac>0$ gives two distinct real roots; $\Delta=0$ gives one repeated real root (the curve touches the $x$-axis without crossing); $\Delta<0$ gives no real root at all (the curve misses the axis entirely).

What a parameter changes. With $a$, $b$, $c$ all fixed numbers, $\Delta$ evaluates to a number and the three cases are read off directly. The moment one of $a,b,c$ is an expression in a parameter $p$, $\Delta$ becomes an expression in $p$, and applying a stated condition ($\Delta>0$, $=0$, $<0$, or a union of two of these) means solving an equation or inequality for $p$ — a genuinely different task, whose answer is a range or pair of values rather than a single number.

The leading-coefficient exclusion, stated as an independent rule. Whenever the parameter appears in $a$ — the coefficient of $x^2$ — the value of $p$ making $a=0$ has to be excluded from the final answer, separately from whatever the discriminant condition itself produces, because at that value the equation is linear and the discriminant has nothing to say about it. This rule is triggered purely by where the parameter sits, not by anything the discriminant expression works out to — it applies equally whether $\Delta(p)$ turns out linear or quadratic in $p$ (§3.1 and §3.3 below both need it; §3.2 and §3.4 don't, because in both of those the parameter never reaches the $a$-slot).

The strict/non-strict boundary, tied to wording. "No real roots" and "two distinct real roots" both exclude $\Delta=0$ (a repeated root is still a real root); "at least one real root" and an unqualified "real roots" both include it; "does not cross the $x$-axis" includes it too, since touching a point is not the same act as crossing through it. The lesson's own prequestions and trap-taxonomy already establish this mapping in full and it is not re-derived here — it is simply the rule §3 below applies at each sub-case's final line.

New to this document: the discriminant as a function of the parameter, in general. Write the quadratic's three coefficients as affine expressions in the parameter $p$ — $A=A_0+\lambda p$, $B=B_0+\mu p$, $C=C_0+\nu p$, where $A_0,B_0,C_0,\lambda,\mu,\nu$ are fixed numbers and any of $\lambda,\mu,\nu$ may be zero (meaning the parameter is simply absent from that slot; this covers every sub-case below, including the lesson's own, as a special case). Expanding $\Delta(p)=B(p)^2-4A(p)C(p)$ and collecting powers of $p$:

$$\Delta(p) = (\mu^2-4\lambda\nu)\,p^2 \;+\; (2B_0\mu-4A_0\nu-4\lambda C_0)\,p \;+\; (B_0^2-4A_0C_0)$$

This is always at most a quadratic in $p$ — never higher-degree, since nothing in $A$, $B$ or $C$ goes above degree 1 in $p$ — but whether it is genuinely degree 2 depends entirely on its own leading coefficient, $\mu^2-4\lambda\nu$. That coefficient is non-zero (so $\Delta(p)$ is a genuine quadratic, needing the critical-value-then-sign-test method) whenever $\mu\neq0$ — the parameter reaches the $b$-slot — or whenever $\lambda$ and $\nu$ are both non-zero — the parameter reaches $a$ and $c$ at once (since $B^2$ contributes a $p^2$ term whenever $\mu\ne0$, and $-4AC$ contributes one whenever both $\lambda,\nu\ne0$, by the ordinary rule for multiplying two things each linear in $p$). If the parameter avoids $b$ entirely and touches at most one of $a,c$, that leading coefficient is forced to zero, and $\Delta(p)$ collapses to an ordinary linear expression in $p$ — a mechanically simpler condition to solve, with no critical-value pair and no sign-test step at all. This single formula is what organises every sub-case in §3: which real or illustrative shape a parameter question takes is fully determined by which of $\lambda,\mu,\nu$ are non-zero, and the formula predicts the resulting $\Delta(p)$'s shape correctly in every case checked below.

#3. The real distinct sub-cases, with the evidence for each

#3.1 Parameter in both $a$ and $b$, $c$ fixed — genuine anchor: Jan 2023, Q4 [real]

The real question, worked here on its own numbers rather than the lesson's substituted ones: $kx^2+6kx+5=0$, $k$ a non-zero constant, has no real roots; find the range of $k$. Here $A=k$ ($\lambda=1$), $B=6k$ ($\mu=6$), $C=5$ fixed ($\nu=0$) — so by §2's formula, $\Delta(k)=(36-0)k^2+(0-0-4(1)(5))k+(0-0)=36k^2-20k$, matching a direct hand-expansion of $(6k)^2-4(k)(5)$ exactly. "No real roots" is the strict case: $36k^2-20k<0 \Rightarrow 4k(9k-5)<0$, critical values $k=0$ and $k=\frac59$, negative between them (upward parabola in $k$, since $36>0$), giving $0<k<\frac59$.

The real mark scheme (re-extracted directly from the primary PDF and cross-checked in the lesson's own audit trail) carries this as 4 marks total, structured M1, dM1, A1, A1 — not five, and this exact count matters: an earlier draft of this course's own lesson had mis-split the inequality-forming step into its own accuracy mark, inflating the real total to five, before a bullet-by-bullet audit against the primary mark scheme caught and fixed it. The real M1 credits attempting $b^2-4ac$ with the correct $a,b,c$ and explicitly condones a candidate who squares only the letter $k$ and not the numeral (writing $6k^2$ in place of $(6k)^2=36k^2$) at this stage only. The dM1, dependent on the M1, is for forming the inequality and solving it to a non-zero critical value — one mark, not two. The first A1 is specifically for identifying the upper critical value, $\frac59$, described in the mark scheme's own words as needing to be "not just the value"; the second A1 is the complete, correctly-strict range. Two of the three real, examiner-quoted error patterns for this question live at points the general theory above already explains: squaring only the letter inside $6k$ produces a clean, equally-plausible wrong upper bound (mark scheme condones the attempt but not the resulting number); "many candidates ignored the solution, $k=0$" — the leading- coefficient exclusion described generally in §2, here coinciding with one of $\Delta(k)$'s own two critical values because $\lambda$ and $\mu$ are both non-zero here (a structural coincidence the lesson's own beyond-spec block already proves in full: whenever the parameter reaches both $a$ and $b$ while $c$ stays fixed, $p=0$ is forced to be a root of $\Delta(p)$, since every term of $\Delta(p)$ then carries a factor of $p$). The lesson's own worked-chain models this identical shape with different numbers ($3px^2+12px+4=0$) and is not re-derived here.

#3.2 Parameter reaching the $b$-slot only after a hidden quadratic is formed — genuine anchor: Jan 2019, Q9 [real]

The real question: $\frac3x+5=-2x+c$, $c$ a constant, has no real roots for $x$; find the range of $c$. This is not yet a quadratic — clearing the fraction is a required first step, and it is worth stating precisely where the parameter $c$ ends up once that step is done, because it is easy to misread from the problem's own letter. Multiplying every term by $x$ (valid since $x\neq0$ is implicit in the original fraction) gives $3+5x=-2x^2+cx$, rearranged to $2x^2+(5-c)x+3=0$ — matching the real mark scheme's own quoted intermediate line exactly. In this quadratic, $A=2$ fixed ($\lambda=0$), $C=3$ fixed ($\nu=0$), and $B=5-c$ — the parameter lands in the $b$-slot, not the $c$-slot the problem's own letter might suggest, with $B_0=5$ and $\mu=-1$. By §2's formula: $\Delta(c)=((-1)^2-0)c^2+(2(5)(-1)-0-0)c+(25-24)=c^2-10c+1$ — a genuine quadratic in $c$, because the parameter reached $b$, exactly as §2 predicts. Setting $\Delta(c)<0$: $c=\frac{10\pm\sqrt{100-4}}{2}=5\pm\sqrt{24}=5\pm2\sqrt6$, matching the real mark scheme's own quoted answer character-for-character, and the range (negative between the roots, upward parabola in $c$) is $5-2\sqrt6<c<5+2\sqrt6$.

The real mark scheme carries this as 7 marks — M1 (clear the fraction) + A1 (the quadratic itself) + M1 (attempt $b^2-4ac$) + dM1 (solve to critical values) + A1 ($c=5\pm2\sqrt6$, exact) + M1 (select which side of the critical values the "no real roots" condition actually needs) + A1 (the final range) — structurally richer than 3.1's four, for two specific reasons the general theory doesn't itself supply but the real record does: an extra method mark to derive the quadratic at all, and a second extra method mark to choose the inside region, which 3.1 never needed because there the "outer" root ($k=0$) was already pinned down as excluded by the stem's own non-zero condition rather than by a separate region-selection step. The dominant real, quoted trap here is recognition, not arithmetic: candidates who "failed to realise that they needed to form a quadratic in $x$" before reaching for the discriminant at all — confirmed directly in the Jan 2019 examiner report — and, separately, a documented algebra slip specific to the fraction-clearing step itself, where some candidates multiplied "all terms except $c$" through by $x$, producing an expression the examiner report explicitly names as linear in $c$ and unable to progress further. The lesson's own marked-solution models this identical shape (hidden quadratic, parameter landing in $b$ once formed) with different numbers ($x+\frac4x=c$) and is not re-derived here.

#3.3 Parameter in $a$ only, or in $c$ only — the linear-in-parameter case: no real anchor found [VERIDIAN, illustrative]

By §2's formula, if the parameter avoids $b$ entirely and reaches at most one of $a,c$, $\Delta(p)$'s quadratic term vanishes and the condition to solve is an ordinary linear inequality — no critical-value pair, no sign-test, no risk of picking the wrong side of a parabola. This is mechanically the simplest shape the topic can take, and — checked against every series this course's research has read — no real WMA11 anchor has been found in this exact shape; the lesson's own MCQ touching a parameter-in-$a$-only equation ($kx^2+3x-7=0$) tests only the conceptual leading-coefficient check, never actually solves the resulting linear condition. Two short constructions fill that gap and show it isn't a trivial one:

Parameter in $a$ only. $kx^2+3x-7=0$, $k$ a non-zero constant, has two distinct real roots. Here $\lambda=1,\mu=0,\nu=0$, so $\Delta(k)=9-4(k)(-7)=9+28k$ (matching §2's formula: leading term $0-4(1)(0)=0$, confirming linearity). The condition $9+28k>0$ gives $k>-\frac9{28}$ directly — a plain rearrangement, no factorising needed. But the stem's own $k\neq0$ still has to be applied, and here it lands somewhere the parameter-in-$a$-and-$b$ shape of §3.1 never produces: since $-\frac9{28}$ is itself negative, $k=0$ sits strictly inside the interval $k>-\frac9{28}$, not at either endpoint of it — so the complete answer is two disjoint intervals, $-\frac9{28}<k<0$ or $k>0$, not the single clean strict range every real anchor in this course's evidence base happens to produce. This is the general point the two real anchors alone can't show: the leading-coefficient exclusion and the discriminant condition's own critical values are only guaranteed to coincide when the parameter reaches both $a$ and $b$ (§3.1's own structural argument); when it reaches $a$ alone, the excluded point can fall anywhere inside the range, splitting it into two pieces rather than trimming one edge of it.

Parameter in $c$ only. $x^2+3x-k=0$ has no real roots. Here $\lambda=0,\nu=-1,\mu=0$, so $\Delta(k)=9-4(1)(-k)=9+4k$ (§2's formula: leading term $0-4(0)(-1)=0$, again linear). The condition $9+4k<0$ gives $k<-\frac94$ directly. Since $a=1$ is fixed and never contains the parameter, no exclusion check of any kind applies — this is the mechanically cleanest of the four shapes in this section, a single interval with nothing to check beyond the discriminant condition itself.

#3.4 Parameter in $b$ only — the symmetric quadratic-in-parameter case: no dedicated real anchor beyond a degenerate case already in the lesson [VERIDIAN, illustrative]

By §2's formula, when the parameter reaches $b$ alone (with $a,c$ fixed and $B_0=0$, i.e. $B$ is exactly proportional to $p$, not merely affine in it), the linear-in-$p$ coefficient $2B_0\mu-4A_0\nu-4\lambda C_0$ collapses to zero for a second, independent reason (each of its three terms vanishes: $B_0=0$ kills the first, $\nu=0$ kills the second, $\lambda=0$ kills the third), leaving $\Delta(p)=\mu^2p^2-4A_0C_0$ — an even function of $p$, with no linear term at all, so its two critical values are always a symmetric pair $\pm k$ for some $k$. No real WMA11 anchor in this exact shape has been found. The lesson's own MCQ closest to it — $x-\frac6x=m$, clearing to $x^2-mx-6=0$ — is this shape ($A_0=1,\lambda=0$; $B_0=0,\mu=-1$; $C_0=-6,\nu=0$, giving $\Delta(m)=m^2+24$ by the formula, matching the lesson's own working) but is deliberately built as the one case where the excluded region is empty, since $m^2+24>0$ for every real $m$ — a genuinely useful teaching point about a parameter question not needing a non-trivial answer, but not a demonstration of what an ordinary, non-empty instance of this symmetric shape looks like. A short construction supplies one: $x^2+8tx+1=0$, $t$ a constant, has two distinct real roots. $\Delta(t)=(8t)^2-4(1)(1)=64t^2-4$ (matching §2's formula: $\mu^2=64$, $-4A_0C_0=-4$). The condition $64t^2-4>0$ gives $t^2>\frac1{16}$, i.e. $t<-\frac14$ or $t>\frac14$ — symmetric about zero, as the general argument predicts, and with no leading-coefficient exclusion needed at all, since $a=1$ here never contains the parameter.

#4. Which real-roots wording is actually confirmed on real scripts

Both real anchors above use the identical real-roots case: "has no real roots", the strict $\Delta<0$ form. Neither "two distinct real roots," "at least one real root," nor "does not cross the $x$-axis" has been found worded into a real WMA11 question testing this exact parameter mechanism — those three framings exist in this course's own teaching material (the lesson's own MCQs and prequestions) as legitimate practice of spec 1.4's other two named cases, not because a real paper has been found using them for a parameter question specifically. Spec 1.4 itself covers all three cases without preference, so a paper is free to ask any of them; the honest state of the evidence is that only one of the four common framings has actually been seen doing so, in the two series this course's research has found.

#5. Scope boundary: constraining a parameter versus counting intersections

Two further real WMA11 findings use "the discriminant" but are not this topic, and are named here only to draw the line clearly, not to be worked through (they belong to other lessons' own evidence bases). Jan 2019, Q11(b)–(c) uses a discriminant ($x^2-8x+13=0$, $\Delta=12$) to find the actual irrational $x$-values of a quadratic-meets-cubic intersection — the unknown being solved for is $x$, not a parameter constraining the equation, so no parameter-solving step of the kind §3 covers is involved at all. Jan 2023, Q7(d) records a different real trap in the reverse direction: candidates who had already found a specific numeric value of a constant reached for the discriminant anyway, to no benefit, when the question had already stated how many intersection points existed — a caution about not invoking this topic's machinery when a value is already pinned down, rather than an appearance of the topic itself.

#6. Synthesis: every real exam angle this specific topic has actually been tested from

Two structurally distinct trap families recur across both real anchors: arithmetic that looks clean but is wrong — mis-squaring a compound coefficient, or writing $b^2+4ac$ — which a real script can produce while the rest of the working looks entirely plausible; and a case silently dropped rather than miscalculated — the parameter's own non-zero constraint (3.1) or the requirement to form a quadratic at all before any discriminant step is legitimate (3.2). Neither trap is an arithmetic slip in the ordinary sense; both are about which steps a candidate remembers to perform at all, not how well they perform the ones they do remember.

#7. Honest limits of the evidence

WMA11-verified-facts.md §6 lists this topic among those with "enough verified material for a genuine standalone lesson," on the strength of exactly two clean, fully mark-coded series: Jan 2023 Q4 and Jan 2019 Q9. signal-frequency-and-boundaries.md reports the same finding numerically: of the five WMA11 series with a fully-read examiner report (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025), this exact topic is confirmed present in 2 of 5 series checked — a genuine, repeatable count, not an estimate, and the two further series read for one targeted question each (June 2022 Q5, June 2023 Q1–Q2, per the facts bank's own 2026-09-25 update) were not found to touch this topic in either of the two questions each was read for. Two series (Oct 2022, Jan 2024) are confirmed as genuine mark schemes but were never mined question-by-question, so this topic's presence or absence in them is simply unknown, not counted as an absence. Every sub-case in §3.3 and §3.4 above is explicitly flagged as having no real anchor at all, and no likelihood figure or grade boundary is stated or implied anywhere in this document — the evidence available does not support either, and none is claimed.

#8. What this document deliberately does not repeat

The lesson file's three prequestions, both teach blocks, its mechanism block's own prose derivation, its diagram of $D(p)=144p^2-48p$, its full five-stage worked-chain and chain-drill, its complete marked-solution (all three parts and the common wrong path), its full seven-item trap-taxonomy, its self-explain prompt, its own beyond-spec structural proof for why $p=0$ recurs when $a$ and $b$ share a parameter, its five MCQs, reference card and closing flag all already exist in discriminant-with-a-parameter.ts and are not reconstructed here. This document's purpose was the general algebraic theory that lesson's teaching prose uses but never states as a single formula (§2), the two real anchors worked on their own actual numbers rather than the lesson's deliberately-different ones (§3.1–3.2), and the sub-cases and citation synthesis that sit alongside — not beneath — that existing content.