Solve for x y=x^3-48x+17

Problem

y=x3−48*x+17

Solution

1. Identify the equation as a cubic equation in the form a*x3+b*x2+c*x+d=0 by setting the expression equal to y

x3−48*x+(17−y)=0

2. Apply the cubic formula or Cardano's method, where the equation is in the depressed cubic form x3+p*x+q=0

p=−48

q=17−y

3. Calculate the discriminant of the cubic equation to determine the nature of the roots.

Δ=(q2)/4+(p3)/27

Δ=((17−y)2)/4+((−48)3)/27

Δ=((17−y)2)/4−4096

4. Determine the real solution using Cardano's formula for x

x=√(3,−q/2+√(,Δ()))+√(3,−q/2−√(,Δ()))

5. Substitute the values of p and q into the general solution.

x=√(3,(y−17)/2+√(,((17−y)2)/4−4096))+√(3,(y−17)/2−√(,((17−y)2)/4−4096))

Final Answer

x=√(3,(y−17)/2+√(,((17−y)2)/4−4096))+√(3,(y−17)/2−√(,((17−y)2)/4−4096))

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