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.

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#### Solution

Let the age of one friend be x years.

then the age of the other friend will be (20 - x) years.

4 years ago,

Age of 1st friend = (x - 4) years

Age of 2nd friend = (20 - x - 4) = (16 - x) years

A/q we get that,

(x - 4) (16 - x) = 48

16x - x^{2} - 64 + 4x = 48

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

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

Comparing this equation with ax^{2} + bx + c = 0, we get

a = 1, b = -20 and c = 112

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

= 400 - 448 = -48

b^{2} - 4ac < 0

Therefore, there will be no real solution possible for the equations. Such type of condition doesn't exist.

Concept: Nature of Roots

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