Q.130. Find the value of.

Question 130.
If a^3 + 12ab^2 = 679 and
9a^2b + 12b^3 = 978,
find a^2 - 4ab + b^2.

Note:

The above question posted on YA! was solved by defeNder and me following similar but different routes leading to the same answer.

However, "Falzoon", my valuable contact on YA! made an interesting observation and suggested an elegant method synchronizing our solutions which I have posted as under.

Answer 130.
a^3 + 12ab^2 = 679 ... ( 1 )

9a^2b + 12b^3 = 978
Multiplying by (2/3),
6a^2b + 8b^3 = 652 ... ( 2 )

Subtracting eq. ( 2 ) from ( 1 ),
a^3 - 6a^2b + 12ab^2 - 8b^3 = 27
=> (a - 2b)^3 = 3^3
=> a - 2b = 3 ... ( 3 )

Adding equations ( 1 ) and ( 2 ),
a^3 + 12ab^2 + 6a^2b + 8b^3 = 1331
=> (a + 2b)^3 = (11)^3
=> a + 2b = 11 ... ( 4 )

Solving eqns. ( 3 ) and (4 ),
a = 7 and b = 2
=> a^2 - 4ab + b^2
= (7)^2 - 4(7)(2) + (2)^2
= - 3.

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