Solve each cubic equation using factoring and the quadratic formula. See Example 7. x^3 - 27 = 0

Verified step by step guidance

1. Rewrite the equation as $x^3-27=0$.
2. Recognize that $x^3-27$ is a difference of cubes, which can be factored using the formula $a^3-b^3=(a-b)(a^2+ab+b^2)$.
3. Identify $a=x$ and $b=3$ since $27=3^3$, and apply the formula: $x^3-27=(x-3)(x^2+3x+9)$.
4. Set each factor equal to zero: $x-3=0$ and $x^2+3x+9=0$.
5. Solve $x^2+3x+9=0$ using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=1$, $b=3$, and $c=9$.

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 the structure of cubic equations is essential for solving them, as it allows for the application of various methods, including factoring and the quadratic formula.

Factoring

Factoring involves rewriting an expression as a product of its factors. For cubic equations, this often includes identifying patterns such as the difference of cubes, which can be factored using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). In the given equation x^3 - 27 = 0, recognizing it as a difference of cubes allows for easier solving.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is used to find the roots of quadratic equations (degree two). After factoring a cubic equation, if it reduces to a quadratic form, this formula can be applied to find the remaining roots. It is a crucial tool in algebra for solving equations that cannot be factored easily.