Is the following situation possible? If so,determine their present ages. The sum of the ages of two friends is $20$ years. Four years ago,the product of their ages in years was $48$.
(D) Let the age of one friend be $x$ years.
Age of the other friend will be $(20-x)$ years.
Four years ago,the age of the first friend was $(x-4)$ years.
The age of the second friend was $(20-x-4) = (16-x)$ years.
According to the given condition:
$(x-4)(16-x) = 48$
$16x - x^2 - 64 + 4x = 48$
$-x^2 + 20x - 64 = 48$
$-x^2 + 20x - 112 = 0$
$x^2 - 20x + 112 = 0$
Comparing this equation with the standard form $ax^2 + bx + c = 0$,we get $a=1, b=-20, c=112$.
The discriminant $D = b^2 - 4ac = (-20)^2 - 4(1)(112) = 400 - 448 = -48$.
Since the discriminant $D < 0$,there are no real roots for this quadratic equation.
Therefore,the given situation is not possible.
Age of the other friend will be $(20-x)$ years.
Four years ago,the age of the first friend was $(x-4)$ years.
The age of the second friend was $(20-x-4) = (16-x)$ years.
According to the given condition:
$(x-4)(16-x) = 48$
$16x - x^2 - 64 + 4x = 48$
$-x^2 + 20x - 64 = 48$
$-x^2 + 20x - 112 = 0$
$x^2 - 20x + 112 = 0$
Comparing this equation with the standard form $ax^2 + bx + c = 0$,we get $a=1, b=-20, c=112$.
The discriminant $D = b^2 - 4ac = (-20)^2 - 4(1)(112) = 400 - 448 = -48$.
Since the discriminant $D < 0$,there are no real roots for this quadratic equation.
Therefore,the given situation is not possible.
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