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# algebra

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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
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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
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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