The sum of the squares of two numbers is 8 . The product of the two numbers is 4 . Find the numbers.

Question

asked Apr 23, 2022 162k views

1 Answer

Cubiclewar · Answer 1 · 2022-04-28T08:39:47+0000

Hello there.

First, assume the numbers
x,~y such that they satisties both affirmations:

With these informations, we can set the following equations:

$\begin{center}\align x^2+y^2=8\\ x\cdot y=4\\\end{center}$

Multiply both sides of the second equation by a factor of
:

$2\cdot x\cdot y = 2\cdot 4\\\\\\ 2xy=8~~~~~(2)^(\ast)$

Make
$(1)-(2)^(\ast)$

$x^2+y^2-2xy=8-8\\\\\\ x^2-2xy+y^2=0$

We can rewrite the expression on the left hand side using the binomial expansion in reverse:
(a-b)^2=a^2-2ab+b^2 , such that:

(x-y)^2=0

The square of a number is equal to
if and only if such number is equal to
, thus:

$x-y=0\\\\\\ x=y~~~~~~(3)$

Substituting that information from
(3) in
(2) , we get:

$x\cdot x = 4\\\\\\ x^2=4$

Calculate the square root on both sides of the equation:

$√(x^2)=√(4)\\\\\\ |x|=2\\\\\\ x=\pm~2$

Once again with the information in
(3) , we have that:

$y=\pm~2$

The set of solutions of that satisfies both affirmations is:

$S=\{(x,~y)\in\mathbb{R}^2~|~(x,~y)=(-2,\,-2),~(2,~2)\}$

This is the set we were looking for.