Find $ab$

Let $a + b = X$ , and $a^3 + b^3 = Y$ , then

$(a+ b)^3 = X^3 = a^3 + 3 a^2 b + 3 a b^2 + b^3 = a^3 + b^3 + 3 a b (a + b) = Y + 3 a b X$

hence,

$ab = \dfrac{ X^3 - Y}{3 X }$

Substituting $X = 5.5$ and $Y = 59.125$ gives us $ab = 6.5$ .

Factoring the second equation yields:

(a +b)(a^2 - ab + b^2) = 59.125.

But, since a + b = 5.5, we see:

5.5*(a^2 - ab + b^2) = 59.125, or

5.5 (a^2 + b^2) - 5.5 ab = 59.125 (equation A)

Squaring the first given equation yields:

a^2 + 2*ab + b^2 = 5.5^2, or

(a^2 + b^2) + 2*ab = 5.5^2 (equation B)

Equations A and B now give a system of two linear equations in the two unknowns (a^2 + b^2) and ab. Solving these equations by standard methods gives ab = 6.5