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Let $a$ and $b$ be real numbers such that $a^3 + 3ab^2 = 679$ and $a^3 - 3ab^2 = 615.$ Find $a - b.$

 Jul 21, 2023
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a^3 + 3ab^2 = 679 

 

a^3 - 3ab^2 = 615 

 

We can subtract both of the equations : 

 

(a^3 + 3ab^2 ) - ( a^3 - 3ab^2 )  = 679 - 615 

 

=         6ab^2 = 64

 

 

 = ab^2 = 64/6

 

Now we need to get rid of the fraction part of 64/6, thus a = 1/6 

 

If a = 1/6 then b = ± 8

 

Now we subtract, we have 2 possibilities :

 

1/6 - (-8 ) = 8 + 1/6 

 

Or 

 

1/6 -8 = -47 / 6 

 Jul 22, 2023

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