# 05 - Sum and Difference of Two Numbers Multiplied to Sum and Difference of Their Squares

**Problem**

Find two numbers such that their sum multiplied by the sum of their squares is 5500, and their difference multiplied by the difference of their squares is 352.

**Answer Key**

**Solution**

*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 13*x* - 9*y* = 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*