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 age of one friend be x years. The age of the other friend will be (20 − x) years.

4 years prior, the age of the first friend was (x − 4) years. The age of the second friend was (20 − x − 4), which simplifies to (16 − x) years.

It is given that \( (x − 4) (16 − x) = 48 \). Expanding this yields \( 16x − 64 − x^2 + 4x = 48 \), which further simplifies to \( − x^2 + 20x − 112 = 0 \). Multiplying by -1, we get \( x^2 − 20x + 112 = 0 \).

Comparing this equation with \( ax^2 + bx + c = 0 \), we identify the coefficients: a = 1, b = −20, and c = 112.

The discriminant is calculated as \( b^2 − 4ac = (− 20)^2 − 4 (1) (112) = 400 − 448 = −48 \). Since the discriminant \( b^2 − 4ac<0 \), there are no real roots for this equation. Consequently, this scenario is not possible.