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?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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.

Approved by eNotes Editorial Team
An illustration of the letter 'A' in a speech bubbles

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.

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial