If `alpha, beta` are the roots of the equation `ax^2 + bx +c=0` then the value of `(1+alpha+alpha^2)(1+beta+beta^2)` is

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khadarabad · Answer 1 · 2024-12-21T19:01:42+0000

α + β = -b/a and αβ = c/a

( 1 + α + α² ) ( 1 + β + β² ) = 1 + β + β² + α + αβ + αβ² + α² + α²β + α²β²

= 1 + (α + β) + ( α² + β² ) + αβ + αβ(α + β) + α²β²

= 1 + (α + β) + ( [ α + β ]² - 2αβ ) + αβ + αβ(α + β) + α²β²

= 1 + (α + β) + [ α + β ]² - αβ + αβ(α + β) + α²β²

= 1 - b/a + b²/a² - c/a - (bc)/a² + c²/a²

= ( a² - ab + b²- ca - bc + c²) /a²

= ( a² + b² + c²- ab - bc - ca ) /a²