ATRACTOR

For quadratic equations there is an alternative geometrical construction to test for the existence of real solutions. More precisely, each polynomial \(x^2 + bx + c\) is associated with a circle (called a Carlyle circle) whose diameter is the segment \(AB\), where \(A = (0,1)\) and \(B = (-b,c)\); the zeros of the polynomial are precisely the abscissas of the intersection points of this circle with the \(x\)'s axis, if they exist. To prove this statement, we can assume that points \(A\) and \(B\) are distinct, otherwise the circle is degenerate, does not intersect the \(x\)'s axis and the polynomial is \(x^2 +1\), which has no real zeros. Since \(A \neq B\), the corresponding Carlyle circle has as Cartesian equation \[x(x + b) + (y - 1)(y - c) = 0\] and now all you have to do is intersect it with the line with equation \(y=0\). The abscissas of the points of intersection are therefore the solutions of the equation \(x(x + b) + c = 0.\)

The following figure illustrates the information given by the Carlyle circle about the real zeros of the polynomials \(x^2 + 2x + 1\), \(x^2 - 5x + 6\), \(x^2 + 2x + 2\) and \(x^2 + 1\). Note that, for the second last polynomial, the Carlyle circle exists but does not intersect the \(x\)-axis, while for the last polynomial the circumference reduces to the point \((0,1).\)

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