Let
x and
y = the numbers
$(x + y)(x^2 + y^2) = 5500$ ← Equation (1)
$(x - y)(x^2 - y^2) = 352$ ← Equation (2)
$\dfrac{(x + y)(x^2 + y^2)}{(x - y)(x^2 - y^2)} = \dfrac{5500}{352}$
$\dfrac{(x + y)(x^2 + y^2)}{(x - y)(x - y)(x + y)} = \dfrac{125}{8}$
$\dfrac{x^2 + y^2}{(x - y)^2} = \dfrac{125}{8}$
$8x^2 + 8y^2 = 125(x^2 - 2xy + y^2)$
$117x^2 - 150xy + 117y^2 = 0$
$(13x - 9y)(9x - 13y) = 0$
For 13x - 9y = 0
$y = \frac{13}{9}x$ ← Equation (3)
From Equation (2)
$(x - \frac{13}{9}x)\left[ x^2 - \left( \frac{13}{9}x \right)^2 \right] = 352$
$(-\frac{4}{9}x)(-\frac{88}{81}x^2) = 352$
$(-\frac{4}{9}x)(-\frac{88}{81}x^2) = 352$
$\frac{352}{729}x^3 = 352$
$x^3 = 729$
$x = 9$ answer
From Equation (3)
$y = \frac{13}{9}(9)$
$y = 13$ answer
