Is the following situtation possible ? If so, then determine the present ages of the mother and her daughter in the problem given below. : The sum of the ages of a mother and her daughter is 25 years. Five years ago, the product of their ages was 58.

Answer

To determine whether the situation is possible and to find the present ages of the mother and daughter, we can set up the problem as follows:

### Step 1: Define the Variables
Let:
- \( x \) = present age of the daughter
- \( y \) = present age of the mother

### Step 2: Set Up the Equations
From the problem statement, we have two pieces of information:
1. The sum of their ages is 25 years:
\[ x + y = 25 \quad \text{(Equation 1)} \]
2. Five years ago, the product of their ages was 58:
\[ (x - 5)(y - 5) = 58 \quad \text{(Equation 2)} \]

### Step 3: Expand Equation 2
Expanding Equation 2:
\[ xy - 5x - 5y + 25 = 58 \]
Rearranging gives:
\[ xy - 5x - 5y + 25 - 58 = 0 \]
\[ xy - 5x - 5y - 33 = 0 \quad \text{(Equation 3)} \]

### Step 4: Substitute from Equation 1 into Equation 3
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 25 - x \]
Substituting this into Equation 3:
\[ x(25 - x) - 5x - 5(25 - x) - 33 = 0 \]
Expanding this:
\[ 25x - x^2 - 5x - 125 + 5x - 33 = 0 \]
Simplifying:
\[ -x^2 + 25x - 158 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - 25x + 158 = 0 \quad \text{(Equation 4)} \]

### Step 5: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -25 \), and \( c = 158 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-25)^2 - 4 \cdot 1 \cdot 158 = 625 - 632 = -7 \]
Since the discriminant is negative, there are no real solutions for \( x \).

### Conclusion
Since we found that the discriminant is negative, it indicates that there are no possible present ages for the mother and daughter that satisfy both conditions. Thus, the situation described in the problem is not possible.