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FInd 1/(a - 1) + 1/(b - 1) if a and b are the roots of the equation 2x^2 - 7x + 2 = 11x - 15.

 May 14, 2021
 #1
avatar+32837 
+1

2x^2 -7x+2 = 11x -15

2x^2-18x+17 = 0      Quadratic formula shows   roots = 7.92783  and 1.07217

plugging into   1/(a - 1) + 1/(b - 1)

  = ~ 14

 May 14, 2021
 #2
avatar+22083 
+1

Using the work of Electric Pavlov:

 

If you leave the solutions in radical form, the answer is exactly 14.

 

Solutions:  [ 18 + sqrt(188) ] / 4     --->     [ 18 + 2·sqrt(47) ] 4     --->     [ 9 + sqrt(47) ] / 2

and            [ 18 + sqrt(188) ] / 4     --->     [ 18 - 2·sqrt(47) ] 4     --->      [ 9 - sqrt(47) ] / 2 

 

If  a  =  [ 9 + sqrt(47) ] / 2    then    a - 1  =  [ 7 + sqrt(47) ] / 2    and    1 / (a - 1)  =  2 / [ 7 + sqrt(47) ]

 

If  b  =  [ 9 - sqrt(47) ] / 2    then    b - 1  =  [ 7 - sqrt(47) ] / 2    and    1 / (b - 1)  =  2 / [ 7 - sqrt(47) ]

 

Multiply both of these fractions by the complement of their divisors to get:

 

2[ 7 - sqrt(47) ] / [ 49 - 47 ]     +     2[ 7 + sqrt(47) ] / [ 49 - 47 ] 

 

=     [  14 - 2·sqrt(47) ]  +  [  14 - 2·sqrt(47) ]      /  2

=             28 / 2

=                14

 May 15, 2021

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