Question

Find two numbers such that their product is four and the sum of their squares is seventeen.

Answer 1

The system of equations of two unknowns is formulated and solved.

`\large\displaystyle\text{$\begin{gathered}\sf \bf{ \left\{\begin{matrix} \ x* y = 4 \\ x^2+y^2 = 17 \end{matrix}\right. \ \Longrightarrow \ y=(4)/(x)} \end{gathered}$}`

`\large\displaystyle\text{$\begin{gathered}\sf \bf{ x^2+\left ((4)/(x) \right )^2=17\ \Longrightarrow\ x^4+16=17x^2 } \end{gathered}$}`

`\large\displaystyle\text{$\begin{gathered}\sf \begin{align*} 0&=(x^2)^2-17(x^2)+16\\ =(x^2-16)(x^2-1) \\ \ \ \ \ \ \ \ \ \ \ =(x+4)(x-4)(x+1)(x-1) \end{gathered}$}`

Possible number pairs are 4, 1 and -4, -1 .