Roots of cubic equations

Cubic equations are polynomial equations of the third degree, generally represented by the formula ax³ + bx² + cx + d = 0, where a ≠ 0.
The roots of any cubic equations are the values of x that satisfy the equation, i.e., those which make the equation equal to zero.
Every cubic equation has exactly 3 roots which might be real or complex numbers. However, not all roots are necessarily distinct.
If the cubic equation has 3 distinct real roots, it will cross the x-axis at three points.
If it has one real root and a pair of complex conjugate roots, it will only cross the x-axis once, at the real root.
By using the Factor theorem, one approach for obtaining the roots of a cubic equation is to first find one root, then factor out that root from the equation to obtain a quadratic equation, which can be solved using standard quadratic methods.
If the coefficients (values for a, b, c, and d) are rational and one geometric approach to finding a real root is to use the Rational Root theorem. This theorem suggests that any rational root, p/q, must be such that p is a factor of the intercept (the constant term d) and q is a factor of the leading coefficient (the coefficient of the cubic term, a).
The discriminant of a cubic equation, Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d², determines the nature of the roots. If Δ > 0, the cubic equation has three distinct real roots. If Δ = 0, it has at least two equal real roots. If Δ < 0, it has one real root and two complex conjugate roots.
Cardano’s Method, also known as the Cardano’s Formula, is another well-known method for solving cubic equations, especially for cases where coefficients aren’t rational.
Ensure that you’re comfortable with applying both graphical and algebraic methods when solving or obtaining roots of cubic equations, as both methods may provide valuable insights in certain scenarios.
Regular practice and exposure to a range of cubic equation problems is recommended in order to be able to proficiently handle different cubic equation scenarios. Be sure to understand worked examples in the text and attempt ample practice questions for optimal understanding.
Also consider practicing real-world application problems related to cubic equations to better understand their relevance and usefulness in different contexts. For example, cubic equations can describe the motion of objects under certain conditions.
Lastly, make sure you understand the underlying mathematical principles, as this will improve both your ability to work through problems and your overall understanding of the subject matter.