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Let a and b be the solutions to 7x^2 + x - 5 = -3x^2 - 9x - 4. Compute (a - 4)/(b - 4) + (b - 4)/(a - 4).

 May 31, 2024
 #1
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Simplify as

10x^2 + 10x - 1

ab = -1/10

2ab = -2/10  = -1/5

 

a + b  =  -1      square both sides

a^2 + 2ab + b^2  = 1

a^2 - 1/5 + b^2  = 1

a^2 + b^2  = 1 + 1/5  =  6/5

 

(a -4) / ( b -4)  +  ( b -4) / (a -4)  =     [  (a - 4)^2  + ( b -4)^2] ]  ' [ (a -4) ( b-4) ]  =

 

[ a^2 - 8a +16  + b^2 - 8b + 16 ]  /  [ ab - 4(a + b) +16 ]   =

 

[  (a^2 + b^2) - 8(a + b) + 32 ]  / [ ab - 4 (a + b) + 16 ]  =

 

[ 6/5 - 8 (-1) + 32 ]  [ -1/10 - 4 (-1) + 16 ]  = 

 

[ 6/5 + 40 ]  / [ 20 - 1/10] =   20497 / 25

 

cool cool cool

 May 31, 2024

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