Question

If α,β are the roots of ax2+bx+c=0 then α^3+β^3/α^−3+β^−3=

Answer 1

Step-by-step explanation:

This is the ageless quadratic theory whereby:

Sum of roots: (â + ß) = -b/a ……….(1)

Product of roots: âß = c/a …………….(2)

Then, find equation with roots (â³ + ß³)

Multiply (1) to power of 3:

(â + ß)³ = (-b/a)³ ……….(3)

â³ + 3â²ß + 3âß² + ß³ = -b³/a³ …….(4)

Substitute (2) in (4)

â³ +3âc/a + 3ßc/a + ß³ = -b³/a³

â³ + ß³ + 3c/a(â + ß) = -b³/a³

â³ + ß³ =(3c/a)(b/a)-b³/a³=3bc/a²-b³/â³

Sum of roots: â³ + ß³=(3abc - b³)/a³ ..(5)

Product of roots: â³ß³ = c³/â³ ……… (6)

The equation whose roots are â³ and ß³

(x - â³)(x - ß³) = x² - xß³ - xâ³ + â³ß³

= x² - x(â³ + ß³) + â³ß³ …………(7)

Substitute (5) and (6) in (7):

= x² - x(3abc - b³)/a³ + c³/a³

= a³x² - x(3abc - b³) + c³

The required equation is given by:

a³x² + x(b³ - 3abc) + c³ = 0 (Ans)