Question 4

We will set up a quadratic equation to represent the situation.

Step 1 — Formulate the equation

Let's say the age of the first friend is $x$ years. The sum of their ages is 20 years. So, the age of the second friend is $20 - x$ years. Four years ago, the first friend's age was $x - 4$ years. Four years ago, the second friend's age was $20 - x - 4 = 16 - x$ years. The product of their ages four years ago was 48.

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

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

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

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

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

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

$\boxed{x^2 - 20x + 112 = 0}$

Step 2 — Check for possibility

We have the quadratic equation $x^2 - 20x + 112 = 0$. Here, $a = 1$, $b = -20$, and $c = 112$. We need to find the discriminant, $D = b^2 - 4ac$.

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

$D = 400 - 448$

$\boxed{D = -48}$

Since the discriminant $D$ is -48, which is less than zero, the quadratic equation has no real roots. This means there are no real values for $x$ that satisfy the given conditions. Therefore, this situation is not possible.

Answer

The given situation is not possible.