# P. Luckey's complex root finder

In an article about nomograms for finding complex roots of quadratic equations, P. Luckey describes a special slide rule for the same purpose.

 Solve the equation z² + p z + q = 0 for complex number z = ζ + i η   (ζ and η are real numbers!) This is equivalent to solving two equations: ζ² − η² + p ζ + q = 0 and 2 ζ η + p η = 0 Luckey's slide rule gives the values d = √¼p² − q   † and η = √q − ¼p²

Below you see a drawing of Luckey's slide rule, and its algorithm. Enter values for p and q, press Go and see what happens

z² + z + = 0 Is q to the left of |p| ? ↓yes ↓no Real roots Complex roots | | η = 0 ζ = −½p | | Slide the value of q under η = 0 Slide the value of |p| above 0 | | Read d under |p| Read η above q | | The roots are −½p ± d The roots are −½p ± i η

In the original paper, ζ is used instead of d but this rather confusing: ¼p² − q is the square root of the discriminant-part of the (real) root, but not the complete (real) root, which is −½p ± √¼p² − q