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From the given information:
1. The sum of their ages is 39: x + y = 39
2. The mother is 3 years younger than the square of the daughter's age: y = x^2 - 3
Now, we can substitute the second equation into the first equation:
x + (x^2 - 3) = 39
x + x^2 - 3 = 39
x^2 + x - 3 = 39
x^2 + x - 42 = 0
Now, we can solve this quadratic equation to find the values of x (daughter's age). By solving this equation, we find:
x = 6 or x = -7
Since age cannot be negative, the daughter's age is 6 years old. Now, we can substitute x back into the second equation to find the mother's age:
Therefore, the daughter is 6 years old, and the mother is 33 years old.