Is the following situation possible ? If so, determine their present ages. Sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Answer

To determine whether the situation is possible and to find the present ages of the two friends, we can follow these steps:

### Step 1: Define the Variables
Let the present ages of the two friends be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of their ages is 20 years:
\[ x + y = 20 \quad \text{(Equation 1)} \]
2. Four years ago, the product of their ages was 48:
\[ (x - 4)(y - 4) = 48 \quad \text{(Equation 2)} \]

### Step 3: Express One Variable in Terms of the Other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 20 - x \quad \text{(Substituting into Equation 2)} \]

### Step 4: Substitute into the Second Equation
Now, substitute \( y \) into Equation 2:
\[ (x - 4)((20 - x) - 4) = 48 \]
This simplifies to:
\[ (x - 4)(16 - x) = 48 \]

### Step 5: Expand and Rearrange
Expanding the left side:
\[ 16x - x^2 - 64 + 4x = 48 \]
Combine like terms:
\[ 20x - x^2 - 64 = 48 \]
Rearranging gives:
\[ -x^2 + 20x - 112 = 0 \]
Multiplying through by -1:
\[ x^2 - 20x + 112 = 0 \quad \text{(Equation 3)} \]

### Step 6: Calculate the Discriminant
To determine if the roots are real, we calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -20 \), and \( c = 112 \):
\[ D = (-20)^2 - 4 \cdot 1 \cdot 112 = 400 - 448 = -48 \]

### Step 7: Analyze the Discriminant
Since the discriminant \( D \) is less than 0, there are no real roots for the quadratic equation. This means that the situation described in the problem is not possible.

### Conclusion
Therefore, the answer is:
**This situation cannot be possible.**

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