QUIZ 3. Cool Quadratics

There is a very important fact to solve this. You can create up to \(2\) intersections between two cool quadratics, because subtraction of two quadratics is also a polynomial with max degree \(2\), therefore there should be at max \(2\) intersections between two cool quadratics.

Now let's use mathematical induction approach for this. Suppose you already drawn \(n-1\) cool quadratics on the plane. And you are going to draw new cool quadratics on the plane, then you can make up to \(2n-2\) intersections on the plane.

Essentially, there should be \(2n-1\) newly created areas by new line, regardless of which intersections they create(unless there are duplicated intersections), because each line segment (either between two intersection or open-ended) will create \(1\) new area.

three_convex

Caption

The third cool quadratic(green line) is added to the example from the question page. It makes \(4\) new intersections and create \(5\) new areas.

Formula

The green line is drawn by \(f(x) = -(5x-9)^2 + 8\), which is also a cool quadratic.

Therefore, if answer is \(a_n\) for \(n\) cool quadratics, then \(a_n = a_{n-1} + 2n-1\) and \(a_1 = 2\), which means

\[a_n = 1 + \sum_{i=1}^{n} (2i-1) = 1 + n(n+1) - n = n^2 + 1\]

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