1.

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Answer:

Step-by-step explanation:

f(x)=4x²-5x-1

αβ are ZEROS of f(x)

so αβ=-1/4,α+β=5/4

=========================

(α-β)²=α²+β²-2αβ

=α²+β²+2αβ-4αβ

=(α+β)²-4αβ

=(5/4)2-4*-1/4

=25/16+1=41/16

so α-β=√41/16=√41  /4

==============================

(ii) α²-β²=(α-β)(α+β)

=√41 /4 *5/4

=5√41 / 16

============

(III)

1/α+1/β

=1/αβ(α+β)

=(5/4) /(-1/4)

=-5

=======

)iv)

1/α-1/β

=1/αβ(β-α)

=(-1/4)(-5/4)

=5/16

==============

(V)

α/β-β/α

=1/αβ(α²-β²)

=(1/ (-1/4) ( 5√41 /16) ( from SOLUTION(ii) )

=-4*5√41 /16

= -5√41 /4

=================

(vi)

1/α-1/β-2αβ

=1/αβ(β-α-2α²β²)

Putting the values

=1/ (-1/4)[ -(α-β)-2α²β² ]

=-4 [ -√41  /4-2*(1/4)² ]

=-4( -√41  /4 -1/8)

=-4*8( -2√41 -1)

=32(2√41  +1)

==============

=-4*( -√41  /4

=


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