NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 4

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 4

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.

Given:

The sum of the present ages of two friends is 20 years.
Four years ago, the product of their ages was 48.

To Find:

1. Is the given situation possible?
2. If yes, determine their present ages.

Formula:

WKT, for a quadratic equation of the form ax2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero.
Δ = b2 - 4ac ≥ 0

Solution:

Let,

The present age of the first friend (in years) = x

Since the sum of their ages is 20 years,

The present age of the second friend(in years) = 20 - x

Four years ago:

Age of the first friend (in years) = x - 4

Age of the second friend (in years) = (20 - x - 4) = (16 - x)

Given, four years ago the product of their ages = 48

⇒ (x - 4)(16 - x) = 48

Expand the left side of the equation:

⇒ 16x - x2 - 64 + 4x = 48

Grouping terms on one side of '=' sign we get,

⇒ -x2 + 20x - 64 - 48 = 0

⇒ -x2 + 20x - 112 = 0

Multiply the entire equation by -1 to make the leading coefficient positive:

⇒ x2 - 20x + 112 = 0
Comparing x2 - 20x + 112 = 0 with ax2 + bx + c = 0, we have:

a = 1

b = -20

c = 112

To determine if the situation is possible, we calculate the discriminant (Δ):

WKT, Δ = b2 - 4ac

⇒ Δ = (-20)2 - 4(1)(112)

⇒ Δ = 400 - 448

⇒ Δ = -48


Since Δ = -48, which is less than 0 (Δ < 0), there are no real roots for 'x'.

This implies that there are no real values for the ages that satisfy the given conditions.

Result:

No, the given situation is not possible because the discriminant of the resulting quadratic equation is negative (Δ = -48), which means there are no real solutions for the ages.
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