Determinant form of quadratic equation, 3 variables, second order (nomogram)

Question

I am looking for a determinant for a second order equation so that I can build a nomogram. The equation is simply: $$ x^{2} +2 a x-c = 0 $$

It can also be written in another format (which is more helpful to me), but I am not sure if it can be done in this format: $$ x^{2} +2 a x- \frac{D A^{2}}{D_{0} } = 0 $$

I have looked at another question asked here but I have not been able to apply that logic to these equations.

Thanks in advance folks, any input is appreciated!

Answer 1

Here is one (please excuse the lack of formatting):

|     -2a       1       1  |
|     -c        0       1  |  =   0
|  x^2/(x-1)  x/(x-1)   1  |

The corresponding determinant equation and nomogram for w^2 + uw + v = 0 is treated in this article of mine:

The Lost Art of Nomography

Since the -c scale is linear, you can use N-chart nomogram blocks to calculate DA^2/D_0 (either one block if D_0 is a constant, or two consecutive blocks if D_0 is another variable). The N-chart provides multiplication or division with linear scales on each side, so the existing -c scale is replaced by the final scale of the N-chart combo.

Ron