#### Question: 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.

#### Explanatory Answer

Let the present age of one of the friends be x years.

Given information: Sum of the ages of the two friends is 20.

∴ The present age of the other friend = (20 – x) years.

4 years ago their ages would have been (x – 4) and (20 – x – 4) years

i.e., (x – 4) and (16 – x) years

Given information: Product of their ages 4 years back was 48.

i.e., (x - 4) (16 – x) = 48

16x – x^{2} - 64 + 4x = 48

x^{2} - 20x + 112 = 0

In this equation a = 1, b = -20, c = 112

∴ Discriminant b^{2} - 4ac = (-20)^{2} – 4 × 1 × 112

= 400 – 448 = -48 < 0

Because the discriminant is negative, the equation has __NO real roots__.

The situation of ages of the two friends given in the question is NOT possible.