Question

if alpha and beta are the zeroes of x2-x-4 find out the value of
1) 1/alpha + 1/beta - alpha beta

2) alpha/beta + beta/alpha +2 ( 1/alpha +1/beta) + 3*alpha*beta..
plzz its urgent......

Answer 1

Sol : We have quadratic equation x² - x - 4.

Given α and ß are their zeroes.

We know that,

Sum of roots = - ( coefficient of x )/ coefficient of x²

α + ß = - ( - 1 ) / 1

α + ß = 1 / 1 = 1.

Now,

Product of roots = constant term / coefficient of x²

αß = ( - 4 ) / 1

αß = -4.

1. 1/α + 1/ß - αß

     ß + α
=  -------------  - αß
         αß

By substituting the values of ( α + ß ) and ( αß ),

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

= ( - 1 / 4 ) + 4

      - 1 + 16
=  ----------------
           4

= 15 / 4.

2. α/ß + ß/α + 2 ( 1/α + 1/ß ) + 3αß


    α² + ß² + 2ß + 2α
=  ---------------------------   + 3αß
             αß

      α² + ß² + 2 ( α+ß )
=  -------------------------  + 3αß   ----- eq.1
            αß


Now ,we don't have the value of ( α² + ß² ), so let's find it ,

( α + ß )² = α² + ß² + 2 αß

By substituting the values of ( α + ß ) and αß in above equation,

 ( 1 )² = α² + ß² + 2 ( - 4 )

 1 = α² + ß² - 8 

 α² + ß² = 1 + 8

 α² + ß² = 9

Now by substituting the values of ( α² + ß² ) ,αß and ( α + ß ) in eq.1,

    9 + 2 ( 1 )
= --------------- + 3 ( - 4 )
        -4


     9 + 2
=  --------------  - 12
       -4 


      - 11
=  --------------  - 12
        4

    -11 - 48
= --------------
         4

= -59/4.

                             ________________________

Answer 2

Therefore 1) 15/4

2) -59/4