How to find the value of an expression using zeroes of a polynomial

Expressions in Terms of a, b, and c for Roots of a Quadratic Polynomial

Given a quadratic polynomial: ax² + bx + c, where a ≠ 0

Assume that α and β are the roots of the given quadratic polynomial.

Known identities:
α + β = -b/a
αβ = c/a

i) α² + β²
= (α + β)² - 2αβ
= (-b/a)² - 2(c/a)
= b²/a² - 2c/a

ii) α² + β² + αβ
= (α² + β²) + αβ
= (b²/a² - 2c/a) + c/a
= b²/a² - c/a

iii) α² + β² - 3αβ
= (α² + β²) - 3αβ
= (b²/a² - 2c/a) - 3c/a
= b²/a² - 5c/a

iv) α³ + β³
= (α + β)³ - 3αβ(α + β)
= (-b/a)³ - 3(c/a)(-b/a)
= -b³/a³ + 3bc/a²

v) α²/β + β²/α
= (α³ + β³) / αβ
= [-b³/a³ + 3bc/a²] / (c/a)
= (-b³/a²c) + (3b/a)

vi) α/β + β/α
= (α² + β²) / αβ
= (b²/a² - 2c/a) / (c/a)
= [b²/(a²) - 2c/a] × (a/c)
= b²/(ac) - 2a/c

vii) √(α/β) + √(β/α)
This simplifies to: (α + β) / √(αβ), only if α, β > 0
= (-b/a) / √(c/a), only defined under positive root constraints
Otherwise, this remains an unsimplified symbolic expression.