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Unformatted text preview: Chapter 03.02 Solution of Cubic Equations After reading this chapter, you should be able to: 1. find the exact solution of a general cubic equation. How to Find the Exact Solution of a General Cubic Equation In this chapter, we are going to find the exact solution of a general cubic equation 2 3 = + + + d cx bx ax (1) To find the roots of Equation (1), we first get rid of the quadratic term ( ) 2 x by making the substitution a b y x 3 − = (2) to obtain 3 3 3 2 3 = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − d a b y c a b y b a b y a (3) Expanding Equation (3) and simplifying, we obtain the following equation 3 27 2 3 2 3 2 3 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + a bc a b d y a b c ay (4) Equation (4) is called the depressed cubic since the quadratic term is absent. Having the equation in this form makes it easier to solve for the roots of the cubic equation (Click here to know the history behind solving cubic equations exactly). First, convert the depressed cubic Equation (4) into the form 3 27 2 1 3 1 2 3 2 3 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + a bc a b d a y a b c a y 3 = + + f ey y (5) where ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = a b c a e 3 1 2 03.02.1 03.02.2 Chapter 03.02 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = a bc a b d a f 3 27 2 1 2 3 Now, reduce the above equation using Vieta’s substitution z s z y + = (6) For the time being, the constant is undefined. Substituting into the depressed cubic Equation (5), we get s 3 = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + f z s z e z s z (7) Expanding out and multiplying both sides by , we get 3 z ( ) ( ) 3 3 3 2 3 4 6 = + + + + + + s z e s s fz z e s z (8) Now, let 3 e s − = ( is no longer undefined) to simplify the equation into a triquadratic equation. s 27 3 3 6 = − + e fz z (9) By making one more substitution, , we now have a general quadratic equation which can be solved using the quadratic formula. 3 z w = 27 3 2 = − + e fw w (10) Once you obtain the solution to this quadratic equation, back substitute using the previous substitutions to obtain the roots to the general cubic equation....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.
 Spring '08
 Kaw,A

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