Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{3}-27=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Identify the Equation Type

We are given the equation \(x^3 - 27 = 0\). This is a cubic equation in the form \(x^3 = a\).

Step 2: Setting the Equation Equal to Zero

The equation \(x^3 - 27 = 0\) can be rewritten as \(x^3 = 27\). Here, we need to find the cube root of 27 to solve for \(x\).

Step 3: Finding the Cube Root

The cube root of 27 is found by asking what number, when multiplied by itself three times, gives 27. We recognize that \(3^3 = 27\). Thus, the solution is \(x = 3\).

Step 4: Verify the Solution

To verify, substitute \(x = 3\) back into the original equation. We have \((3)^3 - 27 = 27 - 27 = 0\), which confirms the solution is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra that involves finding the value of variables that make an equation true. In the given problem, the equation is in the form of a cubic equation: \(x^3 - 27 = 0\). To solve this, the goal is to isolate \(x\) to determine its value.
This is achieved by setting the equation equal to zero, observing that the expression \(x^3\) should equal 27 to satisfy the equation \(x^3 - 27 = 0\).

• Simplify and reorganize the equation to isolate the variable.
• Recognize the form of the equation to apply appropriate solving techniques.

In this case, once the equation \(x^3 = 27\) is identified, the next step is finding the cube root of 27 to solve for \(x\).

Cube Root

The cube root is the value that, when used in three multiplications, equals the original number. In our problem, we need to determine the cube root of 27, which means identifying the number that satisfies the equation \(x^3 = 27\).
Here are steps to find the cube root:

• Consider potential values for \(x\) that, when cubed, result in the given number.
• Recognize familiar cubes, as in the equation \(3^3 = 27\).

Thus, we find that the cube root of 27 is 3, meaning \(x = 3\). This simple recognition falls under commonly known cube numbers: 1, 8, 27, 64, etc. Understanding these helps quickly find solutions to similar problems.

Algebraic Verification

Algebraic verification is the process of checking your solution to ensure it satisfies the original equation. After finding \(x = 3\), verification is essential to confirm its accuracy.
To verify:

• Substitute the found value back into the original equation, replacing \(x\).
• Perform the calculations to see if both sides of the equation balance.

For the given cubic equation, substitute \(x = 3\) to get \((3)^3 - 27\). Calculate it step by step: \(3^3 = 27\), and thus \(27 - 27 = 0\), proving the solution is indeed correct. Verifying helps ensure that calculations are precise and the solution is valid, bolstering confidence in solving algebraic equations.