Answer
The required numbers are $4-2\sqrt{2}$ and $4+2\sqrt{2}$.
Work Step by Step
Let the required numbers are $a$ and $b$.
Now, according to the question, we have $a+b=8$ and $a\cdot b=8$.
Substitute $b=8-a$ from first equation to the second equation, to get
$a(8-a)=8$
$\Rightarrow 8a-a^2=8$
$\Rightarrow a^2-8a+8=0$
$\Rightarrow a=\dfrac{-(-8)\pm \sqrt{(-8)^2-4(1)(8)}}{2}$ using quadratic formula
$\Rightarrow a=\dfrac{8\pm \sqrt{64-32}}{2}$
$\Rightarrow a=\dfrac{8\pm \sqrt{32}}{2}$
$\Rightarrow a=\dfrac{8\pm 4\sqrt{2}}{2}$
$\Rightarrow a=4\pm 2\sqrt{2}$
If $a=4+2\sqrt{2}$, then, $b=8-a=4-2\sqrt{2}$ and when $a=4-2\sqrt{2}$, then, $b=8-a=4+2\sqrt{2}$.
Therefore, the required numbers are $4+2\sqrt{2}$ and $4-2\sqrt{2}$.
