Question

(8) Any depressed quartic can be solved via Ferrari's algorithm. For simplicity, we consider a special case which is easier to deal with by assuming p= 0, Thus we seek to solve f(x) +r r o. The algorithm proceeds as follows (a) Rewrite the equation as b) Add 2r2m m2 to both sides. Here m is a constant we have not yet determined. Show this yields 2 + m)2mm2 (c) Now choose m so that the right hand side is a perfect square: This implies that the discriminant in r of this quadratic equation is zero. Show this means m solves (d) Solve for m using Cardano's method. (e) For this value of m, show we have transformed the right-hand side of Equation 0.1 into (m-2%) (f) Viewing Equation 0.1 as of the form A2 B2, and rewriting it as (A- B)(A+B) 0, show that we can rewrite Equation 0.1 as the product of two quadratic equations in use the quadratic formula on both of these to determine all solutions (including complex ones.

Answer 1

9 m 5 64 27 4- 64 27