Q4:

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:

1. The sum of the ages of two friends is $20$ years.

2. Four years ago, the product of their ages was $48$.

To Find:

Determine if the situation is possible, and if so, find the present ages of the two friends.

Step 1: Defining Variables

Let the present age of the first friend be $x$ years.

Since the sum of their ages is $20$ years, the present age of the second friend is $(20 - x)$ years.

Step 2: Formulating the Equation based on the condition "Four years ago"

Age of the first friend four years ago = $(x - 4)$ years.

Age of the second friend four years ago = $(20 - x - 4) = (16 - x)$ years.

According to the problem, the product of these ages is $48$:

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

Step 3: Expanding and Simplifying the Equation

Expanding the left side using the distributive property:

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

$16x - x^2 - 64 + 4x = 48$

$-x^2 + 20x - 64 = 48$

Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$:

$-x^2 + 20x - 64 - 48 = 0$

$-x^2 + 20x - 112 = 0$

Multiplying by $-1$ to make the leading coefficient positive:

$x^2 - 20x + 112 = 0$

Step 4: Checking for the possibility of the situation

To determine if the situation is possible, we calculate the discriminant ($D$) of the quadratic equation $ax^2 + bx + c = 0$, where $a = 1$, $b = -20$, and $c = 112$.

The formula for the discriminant is $D = b^2 - 4ac$.

$D = (-20)^2 - 4(1)(112)$

$D = 400 - 448$

$D = -48$

Step 5: Conclusion based on the Discriminant

[Since the discriminant $D < 0$, the quadratic equation has no real roots.]

Because the roots are not real, it is impossible to find real values for the ages of the friends that satisfy the given conditions.

Final Answer: The situation is not possible.