Part 2/7:
The proof of this formula involves completing the square, a method where the quadratic is manipulated into a form that highlights its roots. This derivation lays the groundwork for understanding the more complex cubic formula that we will discuss.
The Cubic Equation and Its Solutions
The cubic formula deals with equations of the form:
[
ax^3 + bx^2 + cx + d = 0
]
where ( a \neq 0 ). Unlike the quadratic formula, cubic equations can have up to three real solutions. The cubic formula offers a method to find these solutions, represented as:
[
x = \sqrt[3]{-\frac{b}{3a} + \sqrt[3]{\left( \frac{b^2 - 3ac}{3a^2} \right)^2 + \frac{d}{a}}} + \sqrt[3]{-\frac{b}{3a} - \sqrt[3]{\left( \frac{b^2 - 3ac}{3a^2} \right)^2 + \frac{d}{a}}}
]