Find The Value

The solutions are the following;

When you observed the two equations, you will see clearly they are broken pieces of Cube of Trinomials.

From the expression $a^{2}$ - 4ab + $4b^{2}$ , factor it and you will get (a- 2b).

Making it cube of binomial then we have $a^{3}$ - $6a^{2}b$ + $12ab^{2}$ - $8b^{3}$ .

As we can see $a^{3}$ + $12ab^{2}$ = 679 and - $\frac{2}{3}$ $\times$ ( $9a^{2}b$ + $12b^{3}$ ) = -652

Hence $a^{3}$ - $6a^{2}b$ + $12ab^{2}$ - $8b^{3}$ = 679 - 652 = 27

$\sqrt{27}$ = 3.

The result follows. $a^{2}$ - 4ab + $4b^{2}$ = $\boxed{9}$