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113_1_equations

# 113_1_equations - This yields a = 1 3(3 q-p 2 b = 1 27(2 p...

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EE113: Digital Signal Processing Spring 2008 Prof. Mihaela van der Schaar Prepared by Martin Andersen and Hyunggon Park Solution of general quadratic and cubic equations 1. Any quadratic equation can be reduced to the form ax 2 + bx + c = 0 . The solution of this equation is provided by x = - b ± b 2 - 4 ac 2 a . 2. A general cubic equation y 3 + py 2 + qy + r = 0 may be reduced to the depressed cubic form x 3 + ax + b = 0 by substituting
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Unformatted text preview: . This yields a = 1 3 (3 q-p 2 ) , b = 1 27 (2 p 3-9 pq + 27 r ) . Now let A = 3 s-b 2 + r b 2 4 + a 3 27 , B = 3 s-b 2-r b 2 4 + a 3 27 . The solution of the depressed cubic is x = A + B, x =-A + B 2 + A-B 2 √-3 , x =-A + B 2-A-B 2 √-3 , (1) and y = x-p 3 . [Reference: B. P. Lathi, “Signal Processing and Linear Systems”, 1998.]...
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