Alan is x years old. his mother's age is the square of his age. If Alan's father is two years older than his mother and the sum of all three ages is 80, Alan age has to be determined.

Alan's mother is x^2 years old and his father is x^2 + 2.

The sum of their age is x + x^2 + x^2 + 2 = 80

=> 2x^2 + x - 78 = 0

=> 2x^2 - 12x + 13x - 78 = 0

=> 2x(x - 6) + 13(x - 6) = 0

=> (2x + 13)(x - 6) = 0

=> x = 6 and x = -6.5

As the age has to be positive Alan's age is 6.

**Alan is 6 years old.**

Let y=mother's age, let z=father's age

y=x^2

z=x^2+2

x+x^2+x^2+2=80

Group like terms:

2x^2+x+2=80

2x^2+x-78=0

a=2

b=1

c=-78

Using the quadratic equation x=[-b +/- sqrt(b^2-4ac)]/2a

We calculate x=-6.5 and x=6

His age can't be negative so we rule out x=6.5 and the answer is x=6.

**Therefore Alan is 6 years old.**

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