The Carlyle Circle

The Carlyle Circle is a visual tool used in algebra to solve quadratic equations of the form \(x^2 + bx + c = 0\). It involves plotting a circle based on the coefficients \(b\) and \(c\), where the center and radius relate to these values. The points where the circle intersects the horizontal axis correspond to the solutions (roots) of the quadratic. Essentially, it provides a geometric perspective, allowing you to see how the roots relate to the coefficients, and it can help visualize the nature of the solutions—whether they are real or complex.