Problem 66 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \((x-1)^{3}+8=0\)

Step-by-Step Solution

Answer

The real solution is x = -1.

Step 1: Isolate the Cubed Term

Start by isolating the \((x-1)^3\) term in the equation. To do this, subtract 8 from both sides to get \((x-1)^3 = -8\).

Step 2: Take the Cube Root

To solve for \(x\), take the cube root of both sides of the equation. The cube root of \((x-1)^3\) is \(x-1\), and the cube root of \(-8\) is \(-2\). Therefore, \(x-1 = -2\).

Step 3: Solve for x

Add 1 to both sides of the equation obtained in Step 2 to solve for \(x\). This results in \(x = -2 + 1\), which simplifies to \(x = -1\).

Key Concepts

Real Solutions

A real solution in equations refers to results that can be found on the real number line. These solutions do not involve imaginary or complex numbers.
In the context of the given cubic equation \((x-1)^{3}+8=0\), we have found that the real solution for \(x\) is \(-1\).
This is because our solution set doesn't require any complex number arithmetic or imaginary units, making it a real solution.

• Real solutions are practical and applicable in most everyday scenarios.
• They are found and represented on a number line between negative infinity and positive infinity.

Understanding the nature of real solutions helps ensure that you're interpreting your results accurately in a mathematical context.

Cube Root

Cube roots are fundamentals in solving cubic equations. To understand the cube root, note that it reverses the cubing process.
For instance, the cube of \(x\) is \(x^3\), and the cube root \(\sqrt[3]{x^3}\) returns \(x\).
In our problem, we encountered \((x-1)^3 = -8\). By taking the cube root of both sides:

• The cube root of \((x-1)^3\) is \(x-1\).
• The cube root of \(-8\) is \(-2\), since \(-2 \cdot -2 \cdot -2 = -8\).

Understanding cube roots is essential when you seek to reverse the power for finding solutions, particularly when tackling cubic equations.

Equation Solving

Equation solving involves finding the values of \(x\) that satisfy the given equation. Every equation has specific techniques or rules that depend on the equation form.
This understanding is key to approaching equations like the cubic one, \((x-1)^3+8=0\).

• The first step is identifying the operations applied to the variable; in this case, cubing and adding are key operations.
• Each step strategically aims to reverse these operations to unravel \(x\).
• Simple arithmetic, like additions and subtractions, plays a crucial role in isolating or simplifying terms.

Grasping these concepts is critical as this involves reversing operations and maintaining equation balance to reach solutions.

Isolation of Terms

Isolation of terms in equations refers to the process of rearranging the equation to get the variable alone on one side. This is often the first strategic move in solving equations.
For example, in our equation:

• Start by isolating \((x-1)^3\) on one side, by subtracting \(8\) from both sides resulting in \((x-1)^3 = -8\).
• This move simplifies the equation and makes solving for \(x\) more straightforward.

By systematically isolating terms, you're essentially peeling away layers until the solution is revealed.
This method is crucial in solving not just cubic equations but any equation with multiple terms and operations involved.