new way to make quadratic equations easy

[Benjer]

I was surprised to learn that roots for quadratic equations could be solved using a slide rule, described in the manual for the Post Versalog:

Quote:If any quadratic equation is transformed into the form x^2 + Ax + B = 0, the roots or values of the unknown x may be determined by a simple method, using the slide rule scales. We let the correct roots be -x_1 and -x_2. By factoring, (x + x_1)(x + x_2) = 0. The terms -x_1 and -x_2 will be the correct values of x providing the sum x_1 + x_2 = A and the product of x_1 * x_2 = B. An index of the CI scale may be set opposite the number B on the D scale. With the slide in this position, no matter where the hairline is set, the product of simultaneous CI and D scale readings or of simultaneous CIF and DF scale readings is equal to B. Therefore it is only necessary to move the hairline to a position such that the sum of the simultaneous CI and D scale readings, or the sum of the simultaneous CIF and DF scale readings, is equal to the number A.

As an example, the equation x^2 + 10x + 15 = 0 will be used. We set the left index of CI opposite the number 15 on the D scale. We then move the hairline until the sum of CI and D scale readings, at the hairline, is equal to 10. This occurs when the hairline is set at 1.84 on D, the simultaneous reading on CI being 8.15. The sum x_1 + x_2 = 1.84 + 8.15 = 9.99, sufficiently close to 10 for slide rule accuracy. Roots or values of x are therefore -x_1 = -1.84 and -x_2 = -8.15. Obviously the values of x solving the equation x^2 – 10x + 15 = 0 will be +1.84 and +8.15 since in this case A is negative, equal to -10.

As a second example the equation x^2 – 12.2x - 17.2 = 0 will be used. The left index of CI is set on 17.2 on the D scale. Since this number is actually negative, -17.2, and since it is the product of x_1 and x_2, obviously one root must be positive, the other negative. Also the sum of x_1 and x_2 must equal -12.2. We therefore move the hairline until the sum of simultaneous scale readings is equal to -12.2. This occurs when the hairline is set on 13.5 on the DF scale, the simultaneous reading at the hairline on CIF being 1.275. x_1 is therefore -13.5 and x_2 is 1.275, since x_1 + x_2 = -13.5 +1.275 = -12.225, sufficiently close to -12.2 for slide rule accuracy. The values of x solving the equation are therefore -x_1 = 13.5 and -x_2 = -1.275.