A reminder from eleventh-grade mathematics: a quadratic equation has two real solutions, possibly equal, if its discriminant is non-negative. Yet when reading a problem statement, it is not always easy to spot the quadratic polynomial that will provide the answer.
Here is a little puzzle from everyday life. Working together, Régine and Christophe clean their small apartment in thirty-six minutes. More meticulous than Christophe, Régine takes half an hour longer to do it alone than he does. How long does she take?
Let t be the time in minutes, and let R and C be our two companions' respective "rates"; we obtain 36(R + C) = (t – 30)C, which can also be written as 36(R/C + 1) = (t – 30).
Furthermore, t R = (t – 30)C gives R/C = (1 – 30/t). Hence 36(2 – 30/t) = (t – 30), or
t 2 – 102 t + 1 080 = 0, which has two solutions, 12 and 90. Since t is greater than 30, Régine takes an hour and a half (t = 90) to clean the apartment by herself.
Inscribed and circumscribed circles
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