Comprehensive Study Notes: Quadratic Equations & Newton's Formula (Lecture 04)

Recap: Root sums for a cubic

  • If , ,  are the roots of the cubic equation a x^3 + b x^2 + c x + d = 0, then by Viète’s relations:
    • sum of roots: ++ = -\frac{b}{a},
    • sum of pairwise products: \u0002 + \u0002\u0003 + \u0003\u0001 = \frac{c}{a},
    • product of roots: \u0002\u0003 = -\frac{d}{a}.
  • These relations underpin many cubic-root-sum problems and set up Newton-type identities for power sums.

Newton's Formula: Power sums and recurrences

  • Let nd e the roots of a quadratic ax^2 + bx + c = 0. Define the power-sum sequences for n ≥ 0:
    • S_n = \alpha^n + \beta^n,
    • T_n = \alpha^n - \beta^n,
    • U_n = \alpha^n + q\beta^n, (a generalized form, with parameter q)
    • V_n = (\alpha\beta)^n.
  • For any n ≥ 0 the sequences satisfy the same linear recurrence coming from the polynomial:
    • Quadratic recurrence (monic form a=1 would be simpler; here we keep a in the formula):
      a S{n+2} + b S{n+1} + c Sn = 0, a T{n+2} + b T{n+1} + c Tn = 0,
      a U{n+2} + b U{n+1} + c Un = 0, a V{n+2} + b V{n+1} + c Vn = 0.
  • Initial values (examples):
    • S0 = 2, S1 = \alpha + \beta = -\frac{b}{a}.
  • For higher-degree polynomials (Newton sums), for a monic cubic with roots α, β, γ one has:
    • S_1 = \alpha+\beta+\gamma = -a,
    • S_2 = \alpha^2+\beta^2+\gamma^2 = a^2 - 2b,
    • Recurrence (Newton sums) for k ≥ 0:
      S{k+3} - (\alpha+\beta+\gamma) S{k+2} + (\alpha\beta+\beta\gamma+\gamma\alpha) S{k+1} - \alpha\beta\gamma \, Sk = 0.
      Equivalently, for the cubic ax^3 + bx^2 + cx + d = 0 (monic form x^3 + Ax^2 + Bx + C = 0 is easier):
      S{k+3} + A S{k+2} + B S{k+1} + C Sk = 0 ext{ for } k \ge 0.
  • Important special case for a cubic in standard form x^3 - 9x^2 + 5x - 1 = 0:
    • Here: α+β+γ = 9, αβ+βγ+γα = 5, αβγ = 1.
    • Power sums satisfy: S{n+3} = 9 S{n+2} - 5 S{n+1} + Sn.
    • Initial values: S0 = 3, ag{S0} \ S1 = 9, ag{S1} \ S_2 = 71.
    • Then for n=0: S3 = 9\cdot S2 - 5\cdot S1 + S0 = 9\cdot 71 - 5\cdot 9 + 3 = 597.
    • This matches the example where the given cubic has S1=9, S2=71, S3=597.

Worked example: Quadratic power sums

  • Given the quadratic: 2x^2 - 7x - 3 = 0,\quad a=2, b=-7, c=-3
  • Compute power sums:
    • S_0 = 2,
    • S_1 = -\frac{b}{a} = -\left(\frac{-7}{2}\right) = \frac{7}{2},
    • Recurrence: 2 S{n+2} - 7 S{n+1} - 3 S_n = 0.
  • Compute successive sums:
    • For n=0: 2 S2 - 7 S1 - 3 S0 = 0 \Rightarrow S2 = \frac{7 S1 + 3 S0}{2} = \frac{7(7/2) + 3\cdot 2}{2} = \frac{49/2 + 6}{2} = \frac{49/2 + 12/2}{2} = \frac{61}{4}.
    • For n=1: 2 S3 - 7 S2 - 3 S1 = 0 \Rightarrow S3 = \frac{7 S2 + 3 S1}{2} = \frac{7(61/4) + 3(7/2)}{2} = \frac{427/4 + 21/2}{2} = \frac{427/4 + 42/4}{2} = \frac{469}{8}.
  • Interpretation: If the roots are α, β, then:
    • S_2 = α^2 + β^2 = 61/4,
    • S_3 = α^3 + β^3 = 469/8.
  • Practical note: These recurrences are quick checks or tools to compute sums of powers without solving for α, β explicitly.

Higher-degree Newton sums: general guidance

  • For a polynomial of degree n with roots α1, α2, …, αn and monic form x^n + a{n-1} x^{n-1} + … + a0 = 0, define power sums Sk = ∑{i=1}^n α_i^k.
  • Newton sums give a recurrence for k ≥ n:
    Sk + a{n-1} S{k-1} + a{n-2} S{k-2} + \cdots + a0 S_{k-n} = 0.
  • The initial values are computed from the first few power sums using Viète’s relations (e.g., S1 = -a{n-1}, S2 = a{n-1}^2 - 2 a_{n-2}, etc.).
  • Example pattern (for quartic and higher): the same idea extends by incorporating all known coefficients to produce a linear recurrence among consecutive power sums.

Selected practice problems (conceptual approach and notes)

  • Practical approach to problems in the transcript often uses:

    • Viète relations to express sums and products of roots directly from coefficients.
    • Newton sums to relate power sums S_k to coefficients for k ≥ n.
    • Recurrence relations for sequences like Sn = α^n + β^n, Tn = α^n - β^n, and related variants (Un, Vn) to avoid solving for roots explicitly.

  • Example technique: sum of powers for quadratic roots

    • If α, β are roots of ax^2 + bx + c = 0, then for n ≥ 0:
      a S{n+2} + b S{n+1} + c S_n = 0,

    S0 = 2, S1 = -\frac{b}{a}.

  • Example technique: sum of terms like α^n + β^n in a cubic context often uses: α+β+γ, αβ+βγ+γα, αβγ and the cubic Newton sum recurrence
    S{k+3} - (α+β+γ) S{k+2} + (αβ+βγ+γα) S{k+1} - αβγ \, Sk = 0.

Practical references from the transcript (concepts and methods)

  • JEE Main 2022: Sum of real roots of an equation involving exponentials can be found by converting to a polynomial in e^x (e.g., set t = e^x) and solving a quadratic in t; then sum the corresponding x-values via logs.
  • JEE Main 2020: Power sums Sn for roots of a quadratic satisfy a simple second-order recurrence; use it to derive relations such as a S{n+2} + b S{n+1} + c Sn = 0.
  • JEE Advanced 2017: For sequences an = p α^n + q β^n where α, β are roots of a quadratic, use the same recurrence to relate different n-values and deduce required expressions, often using the fact that α and β satisfy the quadratic.
  • Other problems in the set routinely exploit: the identity S1 = sum of roots, S2 = sum of squares via (sum)^2 - 2(sum of pairwise products), and the cubic Newton sums for higher powers.

Practical notes on the course content from the transcript

  • The material covers: Recap of Vieta's relations for cubic, Newton's sums for power sums, and generalizations to higher-degree polynomials.
  • It emphasizes fast computation of sums of powers of roots without explicitly solving for the roots.
  • It includes numerous worked examples and multiple-choice practice items (with provided answers) to illustrate the techniques.
  • There is emphasis on recognizing patterns in recurrences and using initial sums (S1, S2, etc.) to bootstrap higher S_n values.

Real-world relevance and implications

  • These techniques underpin numerical methods and spectral theory where eigenvalues (roots) are used to compute power sums and invariants of polynomials without explicit solutions.
  • In engineering and physics, power sums appear when expanding polynomials of matrices or in generating functions, where Newton sums provide efficient computational shortcuts.
  • The method also reinforces algebraic thinking: converting a root-based problem into a recurrence problem on power sums.

Quick reference: key formulas to memorize

  • For cubic ax^3 + bx^2 + cx + d = 0 with roots α, β, γ:
    egin{aligned}
    ext{Sum} &:
    \alpha+\beta+\gamma = -\frac{b}{a}, \
    ext{Sum of pairwise products} &:
    \alpha\beta+\beta\gamma+\gamma\alpha = \frac{c}{a}, \
    ext{Product} &:
    \alpha\beta\gamma = -\frac{d}{a}.
    \end{aligned}
  • Newton sums for cubic (monic form x^3 + Ax^2 + Bx + C = 0):
    S1 + A = 0, S2 + A S1 + 2B = 0, S3 + A S2 + B S1 + 3C = 0,
    \Rightarrow S3 = -A S2 - B S_1 - 3C.
  • General quadratic recurrence (for roots α, β of ax^2 + bx + c = 0):
    a S{n+2} + b S{n+1} + c Sn = 0, S0 = 2,
    S_1 = -\frac{b}{a}.
  • Power-sum recurrence for a quadratic from a and b:
    • For cubic-based sums: S{k+3} - (α+β+γ) S{k+2} + (αβ+βγ+γα) S{k+1} - αβγ \, Sk = 0.