5. Find x, from the equation 2x2-375x+1904=0. Here the first figure of the root is 5. 6. Given x2+7x-1194=0, to find x. Ans. 31.2311099. 7. Given x2+x-60=0, to find x. Ans. 7.26208734. 8. Given x2-21x=214591760730, to find x. CHAPTER IV. YOUNG'S SOLUTION OF CUBICS. (Art. 174.) Let x3+Ax2+Bx-N represent a general cubic equation. Conceive the first figure of the root found by trial, as in quadratics, (Art. 173.), and represented by r, and as r is nearly equal to x, Hence, r= find r by trial. r3+Ar2+Br=N, nearly. N r2+Ar+B' nearly; and from this, we As in (Art. 173.) let y represent the remaining part of the root; then x=r+y, or y+ r =x. By+Br =Bx Ay2+2Ary+Ar2=Ax2 y3+3ry2+3 ray+ r3= x3 y3+A'y3+ By + a =N (1) Equation (1) is similar to the primitive equation, and if s represents the first figure of its root, and z the remaining part of its root, we shall have And z3+A"z+B"z=N", &c.; that is, equation after equation will appear in the same form until the root terminates. The successive figures of the roots of the primitive equa tion can therefore be represented as follows: S= N' N' 82+A's+B's(s+3r+A)+3r2+2rA+B N" t=p+A"t+B" t[t+3(+3)+A]+3s3+2A's+B' t2 N" (3) &c. u2+A'"'u+Bu[u+3(r+s+t)+A]+3t2+2A"t+B. (4) N''' &c. &c. EXAMPLES. 1. Given x2+2x2+3x-13089030, to find x. By trial, we soon find that a must be more than 200 and less than 300. Therefore, r-200, A-2, B=3, and N=13089030. 3r2+2rA+B=B'.. 120803 1st trial divisor. s(s+3r+A)=30(632) 18960 2. Given x2+173x=14760638046, to find x. Here, A=0, B=173, and we find by trial that x must be between 2000 and 3000; hence, r=2000 3. Given x2+2x-23x=70, to find an approximate value of x. Ans. x 5.134578. N. B. After the fourth divisor is obtained, it may be used as a divisor in decimals; the remainder being the dividend, the quotient figures will be the true figures of the root for five or six places. |