## Task 3 - 020723

Emil tells: "My brother Heinz is only half as old as I am. If you multiply the number of his years by itself, you get my father's age. My mother is 5 years younger than my father. All together, we are 85 years old."

How old is Emil? Describe how you found the solution!
Let Emil's age in years be denoted by $a$. Then his brother Heinz is $\frac{a}{2}$, his father is $\frac{a^{2}}{4}$, and his mother is $\frac{a^{2}}{4}-5$ years old. Together, they are

$$
a+\frac{a}{2}+\frac{a^{2}}{4}+\frac{a^{4}}{4}-5=85
$$

The equation can be rearranged to $a^{2}+3 a=180$. Using the quadratic formula or by systematic trial ( $a$ must be a divisor of 180), we arrive at $a=12$, so Emil is 12 years old.