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The lighthouse keeper's daughter

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A lighthouse keeper's age today, in whole years, is exactly the square of his daughter's age in whole years. In $9$ years he will be exactly $3$ times as old as she will be then. How old is the keeper today, in years?

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Hint

Call the daughter's age $d$ ; the condition gives a quadratic with only one positive root.

Solution

Today the daughter is $d$ and the keeper is $d^{2}$ .
In $9$ years the condition reads $d^{2} + 9 = 3 (d + 9)$ , that is $d^{2} - 3 d - 18 = 0$ .
This factors as $(d - 6) (d + 3) = 0$ , and only $d = 6$ is a positive age.
So the keeper is $6^{2} = 36$ today, and indeed in $9$ years $45 = 3 \times 15$ .
The answer is $36$ .

Answer: $36$

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