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

Verified step by step guidance

Recognize that the equation $x^3 - 27 = 0$ is a difference of cubes, since $27$ can be written as \$3^3$.

Recall the difference of cubes factoring formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Apply the formula with $a = x$ and $b = 3$ to factor the equation as $(x - 3)(x^2 + 3x + 9) = 0$.

Set each factor equal to zero: $x - 3 = 0$ and $x^2 + 3x + 9 = 0$.

Solve the linear equation $x - 3 = 0$ for $x$, and then solve the quadratic equation $x^2 + 3x + 9 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=3$, and $c=9$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Cubic Equations

Factoring cubic equations involves expressing the cubic polynomial as a product of simpler polynomials, often starting with recognizing special forms like the difference of cubes. For example, x³ - 27 can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²). This step simplifies solving the equation by breaking it into lower-degree factors.

Difference of Cubes Formula

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). This identity helps factor expressions like x³ - 27 by identifying a = x and b = 3. Applying this formula transforms the cubic into a linear factor and a quadratic factor, making it easier to find all roots of the equation.

Quadratic Formula

The quadratic formula, x = [-b ± √(b² - 4ac)] / (2a), is used to solve quadratic equations that cannot be factored easily. After factoring the cubic into a linear and quadratic factor, the quadratic formula helps find the roots of the quadratic part, ensuring all solutions to the original cubic equation are found.

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