Answer: Let's represent the daughter's age as x years. According to the problem, the mother's age is 3 years older than the square of the daughter's age, so the mother's age is (x^2 + 3) years.
Now, we are given that the sum of their ages is 39. So, we can set up the equation:
x (daughter's age) + (x^2 + 3) (mother's age) = 39
Now, we can solve this equation to find the values of x (daughter's age) and (x^2 + 3) (mother's age).
x + x^2 + 3 = 39
Combine like terms:
x^2 + x + 3 - 39 = 0
Now, let's solve this quadratic equation:
x^2 + x - 36 = 0
This equation can be factored as:
(x + 9)(x - 4) = 0
Setting each factor to zero and solving for x:
x + 9 = 0 --> x = -9 (we discard this solution as age cannot be negative)
x - 4 = 0 --> x = 4
So, the daughter's age (x) is 4 years old.
Now, to find the mother's age, we can use the earlier relationship we established:
Mother's age = x^2 + 3 = 4^2 + 3 = 16 + 3 = 19 years old.
Therefore, the daughter is 4 years old, and the mother is 19 years old.