# 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, how old is Alan?

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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.**

Let alan be X

Mom = X^2

Dad X^2+2

If the addition of all three ages are 80 equate the above to 80

X+X^2+X^2+2=80

So when u simplify u end up getting

X+2X^2=78

Then take all values to one side and then u are left with a quadratic equation. Thengo on solving the quadratic equation using the below formula :-

X+2X^2-78=0

Formula :- (b+/-(b^2-4ac))/2a)

U will end up getting two answers as X= 6 and X=-6.5

As age cant have a minus figure the final answer is 6.

So Alns age is 6, mothers age is 36 and the fathers age is 38.