Problem 33 - College Algebra Essentials

Question

Solve each equation in Exercises \(15-34\) by the square root property. $$ (3 x-4)^{2}=8 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{4 + 2\sqrt{2}}{3}\) and \(x = \frac{4 - 2\sqrt{2}}{3}\)

Step 1: Simplify the Equation

The first step is to simplify the equation so that it is in the form \(a^{2} = b\). The current equation is \((3 x-4)^{2}=8\). We don't need to do anything here because the equation is already in the right form.

Step 2: Apply the Square Root Property

The Square Root Property states that if \(x^{2} = a\), then \(x = \sqrt{a}\) or \(x = -\sqrt{a}\). So, apply the square root property, giving us \(3x - 4 = \sqrt{8}\) or \(3x - 4 = - \sqrt{8}\). Remember, \(\sqrt{8}\) simplifies to \(2\sqrt{2}\).

Step 3: Isolate x

To isolate \(x\), add 4 to both sides of the equations from step 2. This gives \(3x = 4 + 2\sqrt{2}\) or \(3x = 4 - 2\sqrt{2}\). To finish isolating \(x\), divide each equation by 3.

Step 4: Final Answer

Doing the last operation from step 3, gives you \(x = \frac{4 + 2\sqrt{2}}{3}\) or \(x = \frac{4 - 2\sqrt{2}}{3}\). These two equations represent the two solutions to the original equation.

Key Concepts

Square Root Property

The Square Root Property is a valuable tool in algebra for solving equations involving squared terms. It's particularly useful when the equation is set up in the form \( a^2 = b \). The property states that if you have \( x^2 = a \), then \( x \) can be either the positive or negative square root of \( a \). Thus, you get two possible solutions: \( x = \sqrt{a} \) and \( x = -\sqrt{a} \).
For example, applying this to the exercise with the equation \( (3x - 4)^2 = 8 \), we take the square root of both sides. This results in two separate equations: \( 3x - 4 = \sqrt{8} \) and \( 3x - 4 = -\sqrt{8} \).
Always remember to consider both the positive and negative roots. This is essential because squaring any number, whether positive or negative, results in a positive value.

Equation Solving

Equation solving is a critical skill in algebra, allowing you to find unknown values that satisfy an equation. The process generally requires manipulation of the original equation to isolate the desired variable.
In our exercise, beginning with \( (3x - 4)^2 = 8 \), solving means breaking down the equation step-by-step using algebraic principles. The aim here is to simplify it and make both sides balance while isolating the variable in question.
Each operation performed must be done equally to both sides of the equation to maintain equality. This includes operations like adding, subtracting, multiplying, or dividing. The core idea is to reverse or undo operations to systematically isolate \( x \).

Simplifying Expressions

Simplifying expressions makes equations easier to solve by reducing them to their most basic form. In algebra, this involves combining like terms, reducing the complexity of radicals, or simplifying coefficients.
During our exercise, the square root \( \sqrt{8} \) is simplified into \( 2\sqrt{2} \), a more manageable form when dealing with further calculations. It's crucial to express terms completely simplified, especially when dealing with roots. This helps in maintaining consistency and preventing errors.
As you simplify, keep an eye out for common factors and like terms that can be combined. The goal is to make the rest of the equation as straightforward as possible for solving subsequent steps.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. The objective is to have the variable of interest on one side of the equation and everything else on the other.
In this particular problem, after applying the Squarer Root Property, we ended up with equations like \( 3x - 4 = 2\sqrt{2} \). The task then is to isolate \( x \) by reversing the steps that involve \( x \).

• Addition or subtraction: Adjust other terms to get the term with the variable alone on one side.
• Division or multiplication: Use these operations to make the coefficient of \( x \) equal to 1.

In our exercise, isolating \( x \) required adding 4 to both sides and then dividing by 3. These steps allow you to find \( x = \frac{4 + 2\sqrt{2}}{3} \) and \( x = \frac{4 - 2\sqrt{2}}{3} \), completing the solution.