Problem 65 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 3(x-3)^{3}=375 $$

Step-by-Step Solution

Answer

The real solution is \( x = 8 \).

Step 1: Isolate the Cubic Expression

Divide both sides of the equation by 3 to isolate the cubic expression. This gives:\[ (x-3)^3 = \frac{375}{3} = 125 \]

Step 2: Apply the Cube Root

Take the cube root of both sides of the equation to solve for the expression inside the cube. This yields:\[ x-3 = \sqrt[3]{125} = 5 \]

Step 3: Solve for x

Add 3 to both sides of the equation to solve for the variable \( x \):\[ x = 5 + 3 = 8 \]

Step 4: Verification (Optional)

Substitute \( x = 8 \) back into the original equation to ensure that it satisfies the equation:\[ 3((8-3)^3) = 3(5^3) = 3(125) = 375 \]The original equation is satisfied, confirming that \( x = 8 \) is correct.

Key Concepts

Isolating the Variable

In equations like the one presented, isolating the variable is an essential first step, particularly when dealing with powers. This means rearranging the equation so that the term involving the variable stands alone on one side of the equation.
To begin, observe the equation: \[ 3(x-3)^{3}=375 \]The term \((x-3)^3\) needs to be isolated. We accomplish this by dividing both sides of the equation by 3:\[ (x-3)^3 = rac{375}{3} = 125 \]With the cubed expression isolated, we have a more straightforward task ahead.
This step is crucial because it simplifies the equation, reducing it to a form where we can subsequently apply simpler operations like taking roots.

• Step ensures clarity on what directly affects the variable.
• Eliminates additional coefficients that may complicate solving the equation.
• Makes the subsequent steps more intuitive and manageable.

Understanding Cube Roots

Taking the cube root of a number involves finding a value that, when multiplied by itself twice more, yields the original number. In this step, we're interested in solving for \( x-3 \) by taking the cube root of 125. This essential mathematical operation helps us to "undo" the cubed expression on an isolated side of the equation:\[ x-3 = \sqrt[3]{125} = 5 \]Knowing the cube root concept is particularly useful when working with cubic equations, as it allows the original problem to be simplified:

• Identifies a form where exponentiation is reversed.
• Transforms a power equation into a linear equation.
• Opens the pathway to find the variable isolated on one side.

Understanding this process is key, as without it, manipulating cubic equations to find simple variable expressions would be a daunting task.

Verification of Solutions

Verification ensures the accuracy of your solution by substituting the value of the variable back into the original equation. This is a step that can confirm our earlier calculations were correct. Once we arrived at \( x = 8 \), we should check to make sure this value satisfies the initial equation:\[ 3((8-3)^3) = 3(5^3) = 3(125) = 375 \]When both sides of the equation are equal, the solution is verified.
This step is optional, yet highly recommended, especially when dealing with homework or exam problems:

• Provides confidence in your solved values.
• Exposes any errors made in prior calculations.
• Acts as a safeguard against missteps in algebraic manipulation.

Making verification a habitual practice can considerably enhance overall problem-solving skills in mathematics.