Question

A father’s age is the square of his son’s age (x). In 20 years’ time the father will be 3 times as old as his son. What are the ages of the father and son?
I am supposed to solve this by writing and solving a quadratic equation but I don't know how. The ages are 64 and 8.

Answer 1

If the current son's age is x, the current age of the father is $x^2$.

In 20 years they will be, respectively, $x+20$ and $x^2+20$ years old, and we know that the father will be 3 times as old as his son:

$x^2+20=3(x+20) \iff x^2+20=3x+60 \iff x^2-3x-40=0$

The quadratic equation has solutions

$x_{1,2}=\dfrac{3\pm\sqrt{9+160}}{2}=\dfrac{3\pm 13}{2}$

So, the two solutions are

$x_1 = \dfrac{16}{2}=8,\quad x_2 = \dfrac{-10}{2}=-5$

We clearly can't accept negative values as solution for a problem involving the age of a person, so the only feasible solution is x=8, and we deduce that the father's age is 8 squared, i.e. 64.