Problem 93 - College Algebra Essentials

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ (3 x-4)^{2}=16 $$

Step-by-Step Solution

Answer

So, the solutions to \((3x - 4)^2 = 16\) are \(x = 2.67\) and \(x = 0\).

Step 1: Remove the square

To get rid of the square on \( (3x - 4)^2 = 16 \), take the square root on both sides. This leads to two possible equations: \(3x - 4 = 4\) and \(3x - 4 = -4\).

Step 2: Solve the first equation

Now, solve the first equation \(3x - 4 = 4\). Add 4 to both side: \(3x = 4 + 4 = 8\). Divide by 3 on both sides, you will obtain \(x = 8/3 = 2.67\) (rounded).

Step 3: Solve the second equation

Next, solve the second equation \(3x - 4 = -4\). Add 4 to both sides: \(3x = -4 + 4 = 0\). Divide by 3 on both sides this time, and you will obtain \(x = 0/3 = 0\)

Step 4: Summary of solutions

Therefore, the solutions to this equation are \(x = 2.67\) and \(x = 0\) which are the roots of the original equation.

Key Concepts

Square Root Method

The square root method is a powerful algebraic tool to solve quadratic equations, especially when they are in the form of complete squares. In the given problem, we have an equation \[(3x - 4)^2 = 16\]. To begin using the square root method, you take the square root of both sides of the equation. This helps in simplifying the equation and reducing the complexity of the problem.

When you take the square root of a number, it breaks down into two possible values: a positive and a negative. For this case, the square root of 16 is 4, and thus our equations become:

• \(3x - 4 = 4\) and
• \(3x - 4 = -4\),

This method converts a squared term into a simpler linear form, allowing for easier solving of quadratic equations. Remember, when dealing with equations involving squares, considering both positive and negative roots is essential for finding all possible solutions.

Equation Solving

Equation solving is the backbone of algebra, involving finding the values of the variable that make the equation true. Let’s look closely at how we solve each equation derived from the square root method.

For the equation \(3x - 4 = 4\), our goal is to isolate the variable \(x\). To do this, add 4 to both sides so that we have:
\(3x = 8\).
Next, to completely isolate \(x\), divide both sides by 3 resulting in:\(x = \frac{8}{3} = 2.67\).

The process is similar for the other equation \(3x - 4 = -4\). Add 4 to both sides to get:
\(3x = 0\).
Finally, divide by 3 to solve for \(x\), leading to:\(x = 0\).

Thus, the solutions for \(x\) based on this method of solving are \(x = 2.67\) and \(x = 0\), which satisfy both equations.

Algebraic Solutions

Algebraic solutions involve mathematical strategies and manipulations to determine the unknowns in an equation. Solving algebraic equations helps in understanding how different elements relate and interact within mathematics.

In our solved example, we used algebraic operations to simplify two linear equations derived from the quadratic equation. This was achieved by performing operations like:

• Addition to both sides to eliminate constant terms.
• Division to isolate the variable term.

Such manipulations are based on the principle that performing the same operation on both sides of an equation keeps the equality balanced and intact. Keeping the equation balanced is crucial for getting correct solutions.

Moreover, once all similar terms are simplified and combined, the variable can be easily isolated, providing a clear solution to the problem. The process highlights why algebra is such a cornerstone in solving real-world and theoretical problems. Understanding these methods of algebraic solutions enhances problem-solving skills significantly.