Solve each cubic equation using factoring and the quadratic formula. See Example 7. x^3 + 64 = 0

Key Concepts

Cubic Equations

Cubic equations are polynomial equations of degree three, typically expressed in the form ax^3 + bx^2 + cx + d = 0. They can have one real root and two complex roots or three real roots. Understanding how to manipulate and solve these equations is crucial for finding their roots, which can be done through various methods, including factoring and applying the quadratic formula.

Factoring

Factoring involves rewriting an expression as a product of its factors, which can simplify solving equations. For cubic equations, this often includes identifying patterns such as the sum of cubes, which can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). Mastery of factoring techniques is essential for breaking down cubic equations into simpler components that can be solved more easily.

Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. In the context of cubic equations, once a cubic is factored into a linear and a quadratic factor, the quadratic formula can be applied to find the roots of the quadratic part. Understanding how to use this formula is vital for solving equations that cannot be factored easily.