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Let a and b be the solutions to 7x^2 + x - 5 = 6x^2 - 5x + 12. Compute 1/(a^2 - 4) + 1/(b^2 - 4).

 May 14, 2024
 #1
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Rearrange as

 

x^2 + 6x - 17 =  0

 

ab = -17

 

2ab = -34

 

a + b  =  -6           square both sides

 

a^2 + 2ab + b^2  = 36

 

a^2 + b^2   - 34 = 36

 

a^2 + b^2  =  70

 

1/(a^2 - 4)  +  1/(b^2 - 4)  =  

 

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

 

[ 70 - 8 ] / [ (-17)^2 - 4(70) +16 ]  =

 

[62] / [ 25]

 

cool cool cool

 May 14, 2024

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