In the equation \( ax^3 + bx^2 + cx + d = 0 \), you are dealing with a cubic equation in the variable \( x \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a \neq 0 \) (since it is a cubic equation). Here's an overview of how to approach solving this type of equation:

### General Strategy for Solving a Cubic Equation:
1. **Factoring (if possible):**
   - Try to find a factor of the form \( (x - p) \), where \( p \) is a root of the equation. This can often be done using trial and error with possible rational roots from the Rational Root Theorem.
   - Once a factor is found, divide the cubic equation by this factor to get a quadratic equation, which can then be solved using the quadratic formula.

2. **Rational Root Theorem:**
   - To find possible rational roots, use the Rational Root Theorem. It suggests that any rational solution will be a factor of \( d \) (the constant term) divided by a factor of \( a \) (the leading coefficient).

3. **Quadratic Method:**
   - After factoring out one root, the remaining part of the cubic equation will be a quadratic equation, which can be solved using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

4. **Cardano's Formula (for non-factorable equations):**
   - For a general cubic equation, there is a more advanced method known as **Cardano's method**, which provides an exact formula for the roots of a cubic equation. It is more complex and involves solving for radicals.

5. **Graphical/Numerical Methods:**
   - If the cubic equation is difficult to solve algebraically, you can also use graphical methods or numerical approximations (like Newton's method) to estimate the roots.

### Example of a Simple Cubic Equation:
If \( x^3 - 6x^2 + 11x - 6 = 0 \), you can factor it as \( (x - 1)(x - 2)(x - 3) = 0 \), giving the roots \( x = 1, 2, 3 \).

Would you like a specific example or further details on a particular solving method?

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### Related Questions:
1. What is the Rational Root Theorem, and how is it applied to cubic equations?
2. How do you use synthetic division to factor cubic polynomials?
3. What are the advantages of Cardano’s method for solving cubics?
4. Can a cubic equation have complex roots? How do you find them?
5. How does the discriminant of a cubic equation determine the nature of its roots?

### Tip:
If a cubic equation has one real root and two complex roots, Cardano's formula can help in finding both types of roots systematically.

Learn how to solve the cubic equation ax³ + bx² + cx + d = 0 using various methods, including factorization, the Rational Root Theorem, and Cardano's formula. This guide is ideal for students in Grades 10-12, covering key concepts such as cubic equations and quadratic formulas.

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