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Let a and b be the solutions to 5x^2 - 11x + 4 = 4x^2 - 17x + 6. Find 1/a^3 + 1/b^3.

 Jul 21, 2024
 #1
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Simplifying the equation, we have x2+6x2=0

In the form mx^2 + nx + p  = 0, the Product of the roots ab = -n/m =   -2  

So 2ab = -4

 

Sum of the roots a + b =  p/m =  -6     

Squaring both sides, we have 


a2+b2+2ab=36a2+b24=36a2+b2=40


1/a3+1/b3=(a3+b3)/(ab)3        Note  :  a^3 + b^3 = (a + b) (a^2 + b^2 - ab)

 

So we have


(a+b)(a2+b2ab)/(ab)3=(6)(402)/(8)=(6)(42/8)=31.5

 

Thus, the answer is 31.5

 

Thanks! :)

 Jul 21, 2024
edited by NotThatSmart  Jul 21, 2024

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