Summer Problem Solving Marathon Solution #14

Dividing by the leading coefficient, we have x² - ⁴⁄₃x + k/3 = 0.

In a quadratic equation in which the x² term has a coefficient of 1, the sum of the roots is the negative of the x coefficient. So the sum of the two roots in our equation is 4/3.

We thus want two real numbers whose sum is 4/3 and whose product is as large as possible. Clearly both must be positive. In fact, we maximize the product by taking each to be half of 4/3. (This is equivalent to the problem of maximizing the area of a rectangle with fixed perimeter, which is done by making the rectangle a square.)

So the two roots are both 2/3. Now we need to find k. In a quadratic equation in which the x² term has a coefficient of 1, the constant term is the product of the roots. So k/3 = (2/3)(2/3), or k/3 = 4/9, or k = 4/3.