SOLUTION: On my last birthday, my friend said to me: "In 15 years' time, your age will be the square of your age 15 years ago!" a) How old am I? b) Could I have said: "In 3 ye

Let your age be A, then
Your age in 15 years time = A+15
Your age 15 years ago = A-15

A + 15 = (A - 15)²
A + 15 = (A - 15)(A - 15)
A + 15 = A² - 30A + 225
     0 = A² - 31A + 210
     0 = (A - 21)(A - 10)

         A - 21 = 0;   A - 10 = 0
              A = 21;       A = 10

Checking A = 21.  in 16 years you will be 36, and 15 years ago you
wer 6, and 36 is the square of 6.

Checking A = 10.  in 15 years you will be 25, and 15 years ago you
lacked 5 years being born, so you were -5, and 25 is the square of -5.
However we don't normally consider yet unborn people to have negative
ages, so we discard A = 10 and say the only solution is A = 21. 

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Let's find out:

Let your age be A, then
Your age in 3 years time = A+3
Your age 15 years ago = A-3

A + 3 = (A - 3)²
A + 3 = (A - 3)(A - 3)
A + 3 = A² - 6A + 9
    0 = A² - 7A + 6
    0 = (A - 6)(A - 1)

         A - 6 = 0;   A - 1 = 0
             A = 6;       A = 1

Checking A = 6.  in 3 years you will be 9, and 3 years ago you
were 3, and 9 is the square of 3.

Checking A = 1.  in 3 years you will be 4, and 4 years ago you
lacked 2 years being born, so you were -2, and 4 is the square of -2.
However we don't normally consider yet unborn people to have negative
ages, so we discard A = 1 and say the only solution is A = 6.

So the answer is "yes", because age 6 checks. 

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Let's find out:

Let your age be A, then
Your age in 5 years time = A+5
Your age 5 years ago = A-5

A + 5 = (A - 5)²
A + 5 = (A - 5)(A - 5)
A + 5 = A² - 10A + 25
    0 = A² - 11A + 20

Unlike the others, this doesn't factor.  So we suspect
there can be no solution unless we allow fractions of
years.   We use the quadratic formula:

x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

x = (-(-11) +- sqrt((-11)^2-4*(1)*(20) ))/(2*(1)) }}}

x = (11 +- sqrt((121-80 ))/2 }}}

x = (11 +- sqrt((41 ))/2 }}}

The two solutions are approximately 8.7 and 2.3.

Checking A = 8.7, approximately.  in 5 years you will be approximately
13.7, and 5 years ago you were approximately 3.7, and 13.7 is approximately
the square of 3.7 (The exact square of 3.7 is 13.69, so it approzimately
works).  However we do not normally use fractions of years in ages, so
we should say the answer is "no, because there is no whole number age"

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You weren't asked for this, but in general:

If I am n(n+1)/2 years old then in n(n-1)/2 years' time I will be 
the square of my age n(n-1)/2 years ago".

Edwin