Correct Answer - Option 4 : Not possible

**Given****∶∶**

Sum of the ages of two friends = 20

5 years ago, the product of their ages in years was 64.

**Formula Used****∶**

Concept of Quadratic Equations; Discriminant, D = b^{2 }– 4ac

If D > 0, then quadratic equation has two unequal real roots.

If D = 0, then quadratic equation has two equal real roots.

If D < 0, then quadratic equation has no real roots.

**Calculation****∶**

Let the present age of one friend be x years

Then the present age of other friend be (20 – x) years

Five years ago, age of one friend was (x – 5)

And the age of another friend was (20 – x – 5) = (15 –x) years

According to the question,

(x -5)(15 – x) = 64

15x – x^{2} – 75 + 5x = 64

x^{2} – 20x + 139 = 0

Now use, Discriminant, D = b^{2} – 4ac

Here, a = 1, b = -20, c = 139

So, D = b^{2} – 4ac = (-20)^{2} – 4 × 1 × 139 = 400 – 556 = - 156

D = - 156

So, D < 0

∵ No real roots exist.

**∴**** The given situation is not possible.**