find the quadratic 22 – 4.5637066x=-7.3371094, which gives the following imaginary roots. (2.43185+1.19298 V-1 r = 19. Find one of the roots of 23—2x=5. Ans. x=2.09455148&c. 20. Find one of the roots of 2x: +3x=90. Ans. x=3.41639728&c. 21. Find one of the roots of 23+x2+x=100. Ans. x=4.26442997&c. 22. Find one of the roots of x3+x=500. Ans. x=7.89500828&c. 23. Find one of the roots of x3 +10.x2 +5x=2600. Ans. 11.00679934&c. SOLUTION OF EQUATIONS ABOVE THE THIRD DEGREE. (301.) It is obvious that the above method which we have employed for cubic equations, will apply equally well for equations of a superior degree. By carefully studying the preceding method we shall be able to deduce, for equations of the nth degree, this general RUL E. I. Having found the first figure of the root, multiply it into the first coefficient and add the product to the second coefficient, which sum, multiply by the same figure and add the product to the third coefficient, and this sum must be multiplicd by the same figureand the product added to the fourth coefficient; and so continue to multiply the last result by this same figure and to add the product to the next succeeding coefficient, until the last coefficient is reached, which last sum must be multiplied by the same figure and the product subtracted from the term constituting the righthand member of the equation ; the remainder we will call the .; FIRST DIVIDEND. Again, multiply the first coefficient by the same figure and add the product to the number under the second coefficient, which sum must be multiplied by the same figure and the product added to the term under the third coeficient; and thus we must continue to multiply and add, until we have obtained the second term under the last coefficient, which result we shall call the FIRST TRIAL DIVISOR. Again, multiply the first coefficient by the same figure of the root and add the product to the last term under the second coefficient, which result must be multiplied by the same figure and the product added to the last number under the third coefficient; and thus we must continue to multiply and add until we reach the coefficient next to the last, when we must again begin with , the first coefficient and multiply and add as before, until we reach the n—2th coefficient; then, again, commencing with the first coefficient we must multiply and add until we reach the n-3d coefficient; we must continue this process until we have thus obtained n terms under the second coefficient, n-1 terms under the third coefficient, n-2 terms under the fourth coefficient, n-3 terms under the fifth coefficient, and so of the rest. II. Find the second figure of the root by dividing the FIRST DIVIDEND by the FIRST TRIAL DIVISOR, proceed with this second figure precisely as was done with the first figure, observing to keep the work so that units shall stand under units, tens under tens, ģc. EXAMPLES. 1. Find one of the roots of the equation 374 +3+4x2 +50=375. |