Problem 31 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the square root property. $$(3 x-1)^{2}=12$$

Step-by-Step Solution

Answer

\(x= \frac{2\sqrt{3} + 1}{3}\) or \(x= \frac{-2\sqrt{3} + 1}{3}\)

Step 1: Isolate the squared term

The given equation is \((3x-1)^{2}=12\). Notice that the term \((3x-1)^{2}\) is already isolated.

Step 2: Apply the square root property

To eliminate the square, apply the square root to both sides of the equation. This gives:\[ \sqrt{(3x-1)^{2}} = \pm \sqrt{12} \]

Step 3: Simplify the square root

Simplify the expression obtained by taking the square root: \[3x - 1 = \pm 2\sqrt{3} \]

Step 4: Solve for x

Solve the resulting linear equations: \[\begin{align*} 3x - 1 &= 2\sqrt{3} \ 3x &= 2\sqrt{3} + 1 \ x &= \frac{2\sqrt{3} + 1}{3} \end{align*}\] and \[\begin{align*} 3x - 1 &= -2\sqrt{3} \ 3x &= -2\sqrt{3} + 1 \ x &= \frac{-2\sqrt{3} + 1}{3} \end{align*}\]

Key Concepts

Solving Equations

When solving equations, our goal is to find the value of the unknown variable that makes the equation true. We often start with a complex equation and, through various steps, simplify it to find the solution.

- **Understand the Equation**: First, read the equation thoroughly to understand its components and structure.- **Identify the Strategy**: Choose the best mathematical strategy to solve it. Common strategies include adding or subtracting terms, multiplying or dividing, and applying properties like the square root property.- **Step-by-Step Process**: Follow the procedure, step-by-step, ensuring not to skip any part or make errors.
In the example \( (3x-1)^2 = 12 \), we are dealing with a squared term. Hence, applying the square root property here helps in breaking down the complexity of the equation. Always remember to check your solution by plugging it back into the original equation to verify it's correct.

Isolating Squared Terms

Isolating the squared term is usually the first step when faced with equations involving squares. Here's how you do it:

- **Move Other Terms Away**: Ensure only the squared term is on one side of the equation. Use addition, subtraction, multiplication, or division as required.- **Simplify if Needed**: After isolating, check if any further simplification is necessary.In \((3x - 1)^2 = 12\), the squared term \((3x - 1)^2\) is already isolated, making our next steps easier. If it weren't, we would perform mathematical operations to isolate it.

Linear Equations

Linear equations are equations of the first order. This means they have variables raised to the power of one and graph as straight lines.

- **Identifying Linear Equations**: Look for equations in the form \ ax + b = c \ where all operations are linear (addition, subtraction, multiplication, division). - **Solving Steps**: To solve, isolate the variable (x) usually by performing inverse operations:

• First, undo any addition or subtraction surrounding the variable.
• Next, undo any multiplication or division.

From our example step, \( 3x - 1 = 2\sqrt{3} \), recognize that we handle it as a linear equation. Add 1 to both sides, then divide by 3 to solve for \(x\). Consequently, we have our solutions: \( x = \frac{2\sqrt{3} + 1}{3} \) and \( x = \frac{-2\sqrt{3} + 1}{3} \). Remember, double-check by substituting your solutions back into the original equation to confirm.