# Ex.4.4 Q4 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends are \(20\) years. Four years ago, the product of their age in year was \(48\) years.

## Text Solution

**What is Known?**

i) The sum of the ages of two friends are \(20\) years.

ii) Four years ago, the product of their age in year was \(48\) years.

**What is UnKnown?**

Checking the possibility of the situation and if yes, find the present ages.

**Reasoning:**

Let the age of friend \(1\) be *\(x\)* years.

Then,

i) Age of friend \(2 = 20 \,–\) age of friend \(1= 20 - x\)

ii) Four years ago, age of friend \(1 = x – 4\)

Four years ago, age of friend \(2 = 20 - x - 4\)

Product of their ages:

\((x - 4)(20 - x - {\rm{ }}4) = 48\)

**Steps:**

\[\begin{align}(x - 4)(16 - x)& = 48\\16x - {x^2} - 64 + 4x &= 48\\ - {x^2} + 20x - 64 &= 48\\ - {x^2} + 20x - 64 - 48&= 0\\- {x^2} + 20x - 112 &= 0\\{x^2} - 20x + 112 &= 0\end{align}\]

Let’s find the discriminant: \(b² - 4ac\)

\(a = 1,\, b = -20,\, c = 112\)

\[\begin{align}b^2- 4ac& = ( - 20)^2 - 4(1)(112)\\& = 400 - 448\\& = - 48\\b^2 - 4{ac} &< 0\end{align}\]

Therefore, there are no real roots. So, this situation is not possible.