Problem 48 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$x^{3}-27=0$$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Understand the Equation

The equation given is a cubic equation in the form \(x^3 - 27 = 0\). The goal is to find the value of \(x\) that makes this equation true.

Step 2: Rearrange the Equation

Start by adding 27 to both sides of the equation to isolate \(x^3\) on the left side. The equation becomes \(x^3 = 27\).

Step 3: Take the Cube Root

Since \(x^3 = 27\), solve for \(x\) by taking the cube root of both sides. The cube root of 27 is 3. So, \(x = 3\).

Step 4: Verify the Solution

Check if \(x = 3\) satisfies the original equation. Substitute 3 back into the equation: \(3^3 - 27 = 27 - 27 = 0\). Since the left side equals the right side, \(x = 3\) is indeed a correct solution.

Key Concepts

Finding Roots

Finding roots is about determining the values of a variable that make an equation true. In the context of cubic equations, like the one given, finding the roots involves solving for the unknown variable, which in this case is \(x\). Consider a cubic equation such as \(x^3 - 27 = 0\):

• The term "cubic equation" indicates that the highest degree of \(x\) is three. In general, cubic equations can have up to three real roots.


• Finding the root means locating the value of \(x\) for which the equation equals zero.

A straightforward approach is evaluating component parts, such as constants (like 27 in \(x^3 = 27\)), and performing operations like taking the cube root to find the suitable value of \(x\). Hence, the solution to \(x^3 = 27\) is identifying \(x = 3\) as its root by concluding that the cube of 3 results in 27.

Remember, finding roots requires careful manipulation of the equation, making sure every step reflects a valid mathematical operation to maintain equality. This foundational skill is crucial for solving polynomial equations of all degrees.

Equation Solving

Equation solving is a systematic approach to finding the values that satisfy a given mathematical statement. Here, we delve into solving the equation \(x^3 - 27 = 0\):

• Understand the equation: Recognize its form and structure. Knowing it is cubic tells us that the coefficient of \(x\) is crucial in our calculations.


• Rearrange the equation: Isolate the term with the variable. By adding 27 to both sides, we transform the equation to \(x^3 = 27\).


• Solve the equation: Use appropriate techniques. In this case, taking the cube root on both sides, which allows us to determine that the solution \(x = 3\).

The steps of solving an equation require precision and care, ensuring both sides of the equation are treated equally. This represents the heart of algebra, teaching students the balance needed to find unknowns, vital for tackling more complicated problems.

Algebraic Techniques

Algebraic techniques include the tools and methods used to manipulate equations and expressions. When working with cubic equations, several key techniques come into play:

• Rearranging and isolating terms: This involves manipulating the equation to isolate the variable term, as seen by moving constants or terms around.


• Taking roots: For cubic equations, computing the cube root is a pivotal technique, used to solve the equation by reverting the cubic power back to a linear state.


• Verification: Once a solution is found, plugging it back into the original equation verifies that it satisfies the equation.

These techniques are intertwined in solving not just cubic equations, but also a wide array of algebraic puzzles. Mastering them enables students to flexibly approach diverse algebraic challenges, providing a toolkit for problem-solving across different math domains.