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.

Answer 1

Let the present age of friend 1 be a x years
Given that,
Sum of the ages of two friends = 20 years

⇒ Present age of friend 2 = (20 – x) years 

And also given that, four years ago, the product of their age was 48.
⇒ Age of friend 1 before 4 years = (x – 4) years
And age of friend 2 before 4 years = (20 – x – 4) years = (16 – x) years
Given that,

( − 4)(16 − ) = 48
⇒ 16 − ² − 64 + 4 = 48
⇒ ² − 20 + 112 = 0

Let D be the discriminant of this quadratic equation.
Then, D = (−20)² − 4 × 112 × 1 = 400 − 448 = −48<0
We know that, to have real roots for a quadratic equation that discriminant D must be
greater than or equal to 0 i.e. D ≥ 0

But D< 0 in the above. So, above equation does not have real roots
Hence, the given situation is not possible.