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