The sum of ages of two friends is 20 years. Four years ago the product of their ages was 48. Show that these statements cannot be true.

Let age of one friend be x and another friend be (20 – x) so that their sum is 20

Age four years ago

Age of one friend would have been (x – 4) and another friend would have been (20 – x – 4) which is (16 – x)

Given that four years ago the product of their ages was 48

⇒ (x – 4)(16 – x) = 48

⇒ 16x – x² – 64 + 4x = 48

⇒ – x² – 64 + 20x = 48

⇒ x² + 64 – 20x + 48 = 0

⇒ x² – 20x + 112 = 0

Comparing equation x² – 20x + 112 = 0 with ax² + bx + c = 0 we get

a = 1, b = – 20 and c = 112

Discriminant (D) = b² – 4ac

⇒ D = (– 20)² – 4(1)(112)

⇒ D = 400 – 448

⇒ D = –48

D<0 which means no real values of x

As no real values of x the equation which we formed using given statements cannot be true

Hence the statements cannot be true