609 ' OF ENGINEERING FOBMULÆ.
Algebraic Fokmulæ—continued,
x : n = m : p; then xp = m n ;
: x, or n : x = x : m, then x2 = m n, and x
Equations.
x : n — m
x : n = m : p, then x : m = n : p, and (x ± ri) : n = (m ± p) : p.
Quadratic Equations, x* + a x = b, then x = — ? ± \/b + '
xln + axn — b, then x = - * i 'X/ b + i
s+Js'2-ip , s- Js^-ip
x + y = s, and x y = p, then x = ——--,and^= —
Cubic Equation^ x^ + ax2 + bx+c—0, becomes aij 4- cj = 0,
± a a2' ab 2
it we put = x — -; Oj = 0--; ana cx = c — — 4— a».
3 3 3 27
Cardan’s solution of x% 4- b x + c = 0 is as follows:—
3
+
x
2 ' ‘ x2/ r 2
This rule is correct if b is positive; or if b is negative and
If b is negative and
the equation has three true roots—
æ = - 2
.2 v 2 ’ v 2
If b is negative and > tllt‘ roots «e real but imaginary.
Approximation Fobmulæ. If x1 approximates to x‘2 + ax + b = 0,
æj2 - b
then x (nearly) — ---—■ .
v J 2xr + a
If aij approximates to a;3 + a x% + b x + c = o, then
2 a;,3 + ax,2 — c
» =----------------------------------.
3xi + 2ax! + b
If X! approximates to x* + a x^ + bx‘2 + c x + d = 0, then
3 ai]4 + 2 a asj3 + b ajj2 — d
LI ’ e
Ixf 4- 3axi + 2fo?i+ c
2 R