Question

if alpha and beta are zeros of polynomial √3x^2-8x+4√3, fing the value of 1/alpha+1/beta-2*alpha*beta

Answer 1

so eq  :  p(x) = √3x² - 8x + 4√3
let's zeroes be α and β

sum of roots = α+β = 8/√3

product of roots = αβ = 4√3/√3 = 4

1/α + 1/β - 2αβ ⇒  α+β - 2(αβ)² / αβ = 8/√3 - 2(4)² / 4 =  2/√3 - 8
 
ans : 2/√3 - 8

hope this helps

Answer 2

alpha and beta are the zeros of √3x² -8x +4√3

so, a + ß = 8/√3
a.ß = 4√3/√3 = 4

now ,

1/a + 1/ß -2a.ß

= ( a + ß)/aß - 2aß

= ( 8/√3)/4 -2× 4

= ( 2/√3) - 8

= ( 2 - 8√3)/√3

= (2√3 -24)/3

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2nd method ,
√3x² -8x + 4√3

√3x² -6x -2x +4√3

√3x ( x - 2√3) -2(x -2√3)

( √3x -2)( x -2√3)

hence, 2/√3 and 2√3 are the roots of
given polynomial ,
so,
let alpha = 2/√3
beta = 2√3

1/alpha + 1/beta -2alpha .beta

1/(2/√3) +1/(2√3) -2×2/√3 × 2√3

= √3/2 + √3/6 -8

= (4√3)/6 - 8

= ( 4√3 -48)/6

= (2√3 -24)/3