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Quadratic Equations

Quadratic full marks come from symmetric functions first: write ∑α, ∑αβ, αβ before expanding — then nature of roots, formation of equations, and location of roots.

Portion AlgebraFocus Core Algebra; roots & symmetric functions join many papersTarget 60–90sPortion share 37.2%Trend Falling share
Syllabus: real coefficients, roots–coefficients relations, formation of equations, symmetric functions of roots. Advanced often asks transformed roots (α+1/α) without needing messy expanded polynomials. Quadratic Advanced is Vieta and parameter gates — expand only when forced.

How Advanced asks this

  • Nature of roots / condition on parameter
  • ∑α², ∑1/α, α²+β² type symmetric asks
  • Equation with roots α+1/α, α², etc.
  • MC on location of roots
  • Formation of quadratic from conditions

Traps that leak marks

  • Expanding before Vieta
  • Sign errors in −b/a
  • Location conditions incomplete
  • Ignoring a=0 parameter edge
  • Solving for roots when the ask only needs sum/product
  • Sign conditions for root location without fixing leading coefficient sign
  • Forgetting complex-root conjugate pairs when coefficients are real

01. Do not expand (x−α)(x−β) early. Most asks are already in elementary symmetric form.

02. Discriminant decides nature; for location of roots use sign charts / f(a)f(b)<0 style conditions carefully with leading coefficient.

03. Common process: identify ask (nature / sum-product / new equation / parameter) → write Vieta → manipulate → verify with a numeric root check when possible.

04. Joins: Complex (conjugate roots), Sequences (recurrence), Calculus (graph of parabola).

05. Advanced lane: nature/location of roots under a parameter is a D-and-sign checklist, not a parabola story. Write D≥0, sum/product signs, and f(k) tests in a fixed order.

06. Formation / symmetric-function stems speak ∑α, ∑αβ — translate Board-style ‘find roots’ into Vieta language before expanding.

Official · 2026

Syllabus for this topic

Official syllabus points for Quadratic Equations only — not the full Algebra portion map. Portion hub: Algebra · Algebra analysis.

Syllabus on this page is topic-scoped: Quadratic equations. The parent portion (Algebra) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Quadratic symmetric functions of roots

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for Quadratic.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Quadratic equations

high

Exam focusRoots–coefficients, symmetric functions of roots, and parameter conditions on D.

Syllabus points

  • Fundamental theorem of algebra; quadratic equations; roots–coefficients; symmetric functions of roots

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Quadratic Equations — practise the knowledge bridge, not only the single-chapter drill.

Mathematics2026 · Paper 2 · Q15–Q16linked-numerical

Exponential curves + area

Calculus – Curves+Definite Integration – Area+Trigonometric equations

Knowledge bridge

Step 1: set y = e^{-x} equal to y = e^{-x}(sin x + cos x) and cancel the common exponential to get a trig equation for intersection abscissae α_i. Step 2: order those roots and set up the definite integral of the difference of the two curves between α_1 and α_4 for the enclosed area β. Step 3: simplify the given log expression in β to a clean numerical value.

Why it feels hard

The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.

What to practise

Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2026 · Paper 2 · Q8multi-correct

DE solution + extrema + intersections

Differential Equations+Functions / AOD+Curve intersections

Knowledge bridge

Step 1: solve x y' = y − x³ (homogeneous/linear in y/x) to get the family y = Cx − x³/2. Step 2: impose the given condition to fix C, then use y' = 0 to locate the local max. Step 3: compare monotonicity on (1,2) and count positive roots of f = g.

Why it feels hard

Multi-correct options test DE, calculus and graph reading together; one sign error in the DE kills several options.

What to practise

After every DE, immediately ask: extrema? monotonicity? intersections with a given g?

2026 year review →Mathematics notes →Quick card →Syllabus →

All mixed joins →

Why methods matter

Common methods that change your score

Quadratic Advanced is Vieta language. These methods stop you from solving for roots when the paper only asked a symmetric function — that habit alone saves minutes every paper.

01

Symmetric first

Trains you to speak Advanced’s language (∑, product) instead of Board-style root hunting.

Do thisAlways write ∑ and product before expanding.

Why it works

Matches how Advanced phrases asks.

See it once

α²+β²=(α+β)²−2αβ — substitute Vieta directly.

02

Transform then Vieta

Formation questions become template work — that is bankable time.

Do thisFor roots α+1/α, compute new sum/product, form equation.

Why it works

Avoids solving for α explicitly.

See it once

If αβ=1, α+1/α=α+β. Often product=1 simplifies transforms.

03

Parameter on D

Turns wordy nature conditions into one inequality you can finish cleanly.

Do thisTranslate verbal nature into inequality on D (and sometimes a,c).

Why it works

Clean SC/NUM conditions.

See it once

Equal roots ⇒ D=0. Real roots ⇒ D≥0.

04

Sign conditions for location

Checklist thinking beats vague parabola sketches when options are close.

Do thisCombine D, sum, product, and f(k) signs systematically.

Why it works

Location MC has many near-miss options.

See it once

Both roots >1 needs careful f(1)>0, sum>2, etc. with leading coeff — write a checklist.

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

bankFamily · Symmetric function of roots

Sample question

If α,β are roots of x²−5x+6=0, find α²+β².

Symmetric asks should never start by solving the quadratic unless forced.

Model chosenVieta → (∑)²−2∏.

Processed solution

  1. Vieta: ∑α = 5, αβ = 6 — do not solve for α,β.
  2. Identity: α²+β² = (∑α)² − 2αβ = 25 − 12.
  3. Value = 13.

Answer13

Takeaway: Write ∑ and product before any expansion.

02

standardFamily · Transformed roots → new equation

Sample question

Roots of x²−3x+2=0 are α,β. Form equation with roots α+1, β+1.

Transformed-root formation is a standard Advanced pattern — new ∑/∏ only.

Model chosenNew ∑=∑+2, new ∏=(α+1)(β+1).

Processed solution

  1. ∑=3, ∏=2.
  2. New ∑ = 3+2=5; new ∏ = 2+3+1=6.
  3. Equation: x²−5x+6=0.

Answerx² − 5x + 6 = 0

Takeaway: Never find α,β individually for formation asks.

03

twistFamily · Transformed roots → new equation

Sample question

If α, β are roots of x² − 3x + 1 = 0, form the monic quadratic with roots α/(α − 1) and β/(β − 1).

Advanced formation asks reward Vieta on the transform — solving radicals wastes the budget.

Model chosenNew ∑ and ∏ via symmetric functions only.

Processed solution

  1. Vieta: α + β = 3, αβ = 1.
  2. Denominator (α − 1)(β − 1) = αβ − (α + β) + 1 = 1 − 3 + 1 = −1.
  3. New sum = [α(β − 1) + β(α − 1)] / (−1) = (2αβ − α − β)/(−1) = (2 − 3)/(−1) = 1.
  4. New product = αβ/(−1) = −1.
  5. Equation: t² − t − 1 = 0.

Answert² − t − 1 = 0

Takeaway: Transform ∑ and ∏ — never hunt individual roots for formation.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Symmetric function of roots

Looks likeα²+β², α³+β³, 1/α+1/β, α/β+β/α

Fast modelWrite ∑α, αβ first — never expand (x−α)(x−β) early.

Same solve path

  1. Vieta: ∑ and product.
  2. Apply identity (∑α²=(∑)²−2∏, etc.).
  3. Substitute numbers.

Rough skeleton

  • ∑, ∏
  • Identity
  • Value

Why faster

Advanced asks are already in symmetric language.

Refuse this

Do not solve for α,β individually unless D is tiny and needed.

02

Nature / parameter on D

Looks likeFind λ so roots real/equal/opposite sign

Fast modelTranslate words → inequality on D (and sum/product if sign conditions).

Same solve path

  1. Write D(λ).
  2. Translate condition.
  3. Solve inequality.
  4. Exclude a=0 edge if parameter hits it.

Rough skeleton

  • D=…
  • Condition
  • λ ∈ …

Why faster

One discriminant inequality replaces casework on explicit roots.

Refuse this

Do not use quadratic formula for every λ.

03

Transformed roots → new equation

Looks likeRoots α+1/α, α², (α+β)/(αβ), … form the equation

Fast modelCompute new ∑ and ∏ from old Vieta; write x²−(∑′)x+∏′=0.

Same solve path

  1. Old ∑, ∏.
  2. New ∑′, ∏′.
  3. Monic equation.
  4. Clear denominators if needed.

Rough skeleton

  • Old Vieta
  • New ∑/∏
  • Equation

Why faster

Avoids finding α explicitly then rebuilding.

Refuse this

Do not solve the original quadratic for α unless stuck.

04

Location of roots

Looks likeBoth roots >1, roots in (0,1), one positive one negative

Fast modelChecklist: D, sum, product, and f(k) signs with leading coefficient — not graphs.

Same solve path

  1. List required inequalities.
  2. Check D≥0.
  3. Sum/product signs.
  4. f at boundary if needed.

Rough skeleton

  • D
  • ∑
  • ∏
  • f(k)
  • Verdict

Why faster

Checklist catches incomplete option sets Advanced loves.

Refuse this

Do not rely on vague parabola sketches alone.

Hall timing

60–90s process

  1. 0–15s
    Write ax²+bx+c=0. Circle ask: nature / symmetric / new roots / parameter.

    Note if roots are real/complex restriction. Underline ‘both roots positive’ type location asks.

  2. 15–35s
    Write ∑α=−b/a, αβ=c/a. For nature, write D=b²−4ac.

    This is the default start — not quadratic formula unless individual roots are needed and D is clean.

  3. 35–75s
    Manipulate symmetric identities or form new monic equation from new ∑ and product.

    ∑α²=(∑α)²−2∑αβ. For new roots γ=g(α), compute ∑γ and ∏γ from old symmetric sums.

  4. 75–90s
    Check D sign vs claim; for location, check necessary conditions (sum/product signs).

    Both roots positive ⇒ sum>0, product>0, D≥0 (with a>0 careful). Verify one numeric example if parameter boundary.

After the model is locked

Quick formula bank

Vieta

α+β=−b/a; αβ=c/a

Almost all quadratic stems

Discriminant

D=b²−4ac

Nature of roots

Roots formula

α,β=[−b±√D]/(2a)

Explicit roots needed

∑α²

(α+β)²−2αβ

Symmetric power sums

∑1/α

(α+β)/αβ (αβ≠0)

Reciprocal root transforms

Formation

x²−(sum)x+(product)=0 (monic)

New equation from transformed roots

Condition for common root

Use gcd / resultant style: α satisfies both; eliminate

Two quadratics share a root

Mental bank

  • α+β=−b/a, αβ=c/a
  • D=b²−4ac
  • α²+β²=(α+β)²−2αβ
  • x²−(sum)x+(product)=0
  • Equal roots ⇒ D=0
  • α+β=−b/a, αβ=c/a
  • D=b²−4ac; equal roots ⇔ D=0
  • Both roots >k ⇒ D≥0, sum>2k, f(k)>0 (a>0 case discipline)
  • Equation with roots α²,β² → replace via symmetric polys

Make it automatic

Practice · checklist

Daily

  • 2 symmetric-function NUM.
  • 1 parameter–nature inequality.
  • 1 transformed-roots equation.

Weekly

  • 1 location-of-roots MC set.
  • 1 join with Complex conjugate roots.
  • Error log: Vieta sign slips.

100% checklist

  • Vieta written before expansion.
  • D used correctly for nature.
  • Transformed roots via new ∑/∏.
  • Location checklist complete.
  • ≥90% on last timed Quadratic set.
  • Parameter conditions reduced to inequalities in D and Vieta.
  • Symmetric asks handled without explicit root hunting when possible.
  • Location checklist includes f(boundary) and sum/product signs.
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