To this day, the formula providing the solution to the cubic equation is known a

Question

To this day, the formula providing the solution to the cubic equation is known as Cardano's formula. We consider here Cardano's treatment of one of the thirteen cases of the cubic, the case x3 = cx + d ("cube equal to the thing and number") presented in chapter 12 of the Ars magna. Cardano asserted that the solution is of the form x = u + v, where u v = c/3 and u3 + v3 = d. He demonstrated this geometrically, but his argument can be translated into algebra: x3 = (u + v)3 = u3 + 3 u2v + 3uv2 + v3 = 3 uv(u + v) + u3 + v3 = cx + d. Given this proof, Cardano then showed how to find u and v that satisfy the two relationships: "When the cube of one-third the coefficient of x is not greater than the square of one-half the constant of the equation, subtract the former from the latter and add the square root of the remainder to one-half the constant of the equation and, again, subtract it from the same half.... The sum of the cube roots of [these two quantities] constitutes the value of x."Cardano has therefore provided an algorithm for solving x3 = cx + d. This algorithm can be translated into the formula Use Cardano's formula to solve x3 + 3x = 10.

Explanation / Answer

using the cardno's formula here c = -3 and d = 10 so c/3 = -1 and d/2 = 5...........putting these values in the given equation we get x= cuberoot(5+sqrt(25 +1)) + cuberoot(5-sqrt(25 +1)) = cuberoot(5 - sqrt(26) ) -+ cuberoot(5 + sqrt(26) = 1.698921... === >>> answer ............hope this helps////////

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