Solution:

Let’s say, the age of one friend be x years.

Then, the age of the other friend will be (20 – x) years.

Four years ago,

Age of First friend = (x – 4) years

Age of Second friend = (20 – x – 4) = (16 – x) years

As per the given question, we can write,

$\left( x-4 \right)\text{ }\left( 16-x \right)\text{ }=\text{ }48$

$16x-{{x}^{2}}-64\text{ }+\text{ }4x~=\text{ }48$

$~\text{ }{{x}^{2}}~+~20x-112\text{ }=\text{ }0$

${{x}^{2}}-20x\text{ }+~112\text{ }=\text{ }0$

Comparing the equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get

$a~=~1,~b~=\text{ }-20~and~c~=\text{ }112$

Discriminant $=~{{b}^{2}}-4ac$

$=\text{ }{{(-20)}^{2}}-4\text{ }\times \text{ }112$

$=\text{ }400-448\text{ }=\text{ }-48$

${{b}^{2}}~-4ac~<\text{ }0$

Therefore, there will be no real solution possible for the equations. Hence, condition doesn’t exist.