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Solve

\[ (x+4)^3 + (x+5)^3 = (x+7)^3 + (x-4)^3 . \]

 Jan 14, 2021
 #1
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( x + 4)^3  +  (x + 5)^3  = (x + 7)^3  + ( x - 4)^3

 

In the expansion of this, the  x^3  terms on each side will "cancel"  and we  are left with

 

3x^2(4) + 3x(4)^2  + 4^3  + 3x^2(5) + 3x*(5)^2 + 5^3  =  3x^2(7) + 3x(7)^2 + 7^3 - 3x^2(4) + 3x(4)^2  - 4^3

 

Simplify

 

12x^2  + 48x + 64  + 15x^2  + 75x  +125  =  21x^2  + 147x  + 343 - 12x^2 + 48x  - 64

 

27x^2  + 123x + 189 =  9x^2 + 195x + 279 

 

18x^2  - 72x  - 90  =  0       divide through by  18

 

x^2  - 4x -  5   =   0       factor

 

(x - 5) ( x + 1)   =   0  

 

Set each factor to 0  and solve for x and we  get that

 

x =  5       or   x =   -1

 

 

cool cool cool

 Jan 14, 2021

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