Part 4/5:
After resolving and collecting like terms, the equation can be reduced to a standard quadratic form, enabling the application of the quadratic formula.
Deriving the Quadratic Formula
To solve the quadratic equation, we recall the standard form:
[
x^2 + Px + Q = 0
]
Using the quadratic formula yields:
[
x = \frac{-P \pm \sqrt{P^2 - 4Q}}{2}
]
In our case, the variables convert so that ( z^3 = -\frac{p}{2} ), ( P = \frac{p^2}{4} ), and ( Q = \frac{p^3}{27} ). By substituting these values back, we can clearly express ( z^3 ) in terms of ( p ) and ( Q ), leading us to the final roots of the original cubic equation.