Part 2/5:
we will manipulate this equation by substituting (x) with a new variable adjustment, ultimately transitioning to a form manageable with quadratic formulas. This approach opens the door to simplification and ultimately solving the cubic equation using techniques similar to those employed in quadratic equations.
Applying the PQ Substitution Method
Step 1: Restructuring the Variable
We initiate this process by letting:
[
x = y + k
]
This substitution is aimed at removing the (x^2) term. After applying this substitution to the original cubic equation, we expand it leading us to new expressions for each term—especially focusing on how to set (k) in a way that ensures the elimination of (y^2).
Step 2: Choosing (k) to Cancel Out (y^2)