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.
No
Step 1: Represent the present ages.
Let the present age of one friend be \(x\) years.
Since their sum is 20 years, the present age of the other friend is:
\[20 - x\]
Step 2: Write their ages four years ago.
Four years ago, each friend was 4 years younger, so their ages then were:
First friend: \(x - 4\)
Second friend: \((20 - x) - 4 = 16 - x\)
Step 3: Use the product condition from four years ago.
We are told that four years ago, the product of their ages was 48. Therefore:
\[(x - 4)(16 - x) = 48\]
Step 4: Expand and simplify the equation.
Expand the left-hand side:
\[(x - 4)(16 - x) = x · 16 - x^2 - 64 + 4x = -x^2 + 20x - 64\]
So the equation becomes:
\[-x^2 + 20x - 64 = 48\]
Move 48 to the left-hand side:
\[-x^2 + 20x - 64 - 48 = 0\]
\[-x^2 + 20x - 112 = 0\]
Multiply the whole equation by \(-1\) to make the coefficient of \(x^2\) positive:
\[x^2 - 20x + 112 = 0\]
Step 5: Use the discriminant to check for real solutions.
For \(ax^2 + bx + c = 0\), the discriminant is \(D = b^2 - 4ac\).
Here, \(a = 1\), \(b = -20\), \(c = 112\).
Compute the discriminant:
\[D = (-20)^2 - 4(1)(112) = 400 - 448 = -48\]
Step 6: Interpret the discriminant.
Since \(D = -48 < 0\), the quadratic equation has no real roots.
This means there is no real value of \(x\) that satisfies the given conditions for the ages.
Conclusion: The situation described is not possible; there are no real ages that meet both the sum and product conditions.