Question

X²+2x+3=0. a=1 b=2 c=3.
1).alfa²beta+beta²alfa=? 2).alfa²-beta²=?
3).1/alfa-2 + 1/beta-2=? 4). (alfa+beta)²-(alfa-beta)²=?

Answer 1

Α+ β = -b/a = -2
α β = c/a = 3

1) α²β +αβ² = αβ(α+β) = 3(-2)  = - 6
2) α² - β² = (α+β)(α-β) = (α+β) √(α-β)² = (α+β) √[α²+b² - 2 ab]
= (α+β) √[(a+b)² - 4αβ]
= (-2) √[(-2)² -4(3)]
= (-2)√(4 -12)
= -2√(-8) ?
3) 1/(α-2) +1/(β-2) = (α-2)+(β-2) /(α-2)(β-2) 
= [(α+β) - 4 ]/(αβ-2(α+β) 
= [-2 - 4 ] / [3 - 2(-2)]
= (-6) /(3+4)
= - 6/7

4) (α+β)² - (α-β)² =
= (α+β)² - [ α²+β² - 2αβ]
= (α+β)² - [(α+β)² - 4αβ]
= (-2)² - [(-2)² -4(3)]
= 4 - [4-12]
= 4 - (-8)
= 4 + 8
= 12
..