Problem 22 - Algebra for College Students

Question

Use the even-root property to solve each equation. $$(a-2)^{2}=8$$

Step-by-Step Solution

Answer

\(a = 2 \text{±} 2√2\).

Step 1: Understand the Even-Root Property

The even-root property states that if \( x^2 = k \), then \( x = \boxed{\text{±}\text{\text{s}}\text{r}t{k}}\). In this case, \((a-2)^2 = 8\).

Step 2: Apply the Square Root

Take the square root of both sides of the equation: \(\text{\text{s}}\text{r}t{(a-2)^2} = \text{\text{±}\text{s}}\text{rt}{8}\). This simplifies to:\(|a-2| = \boxed{\text{2√2}}\).

Step 3: Separate into Two Equations

Since \(|a-2| = 2√2\), we can create two separate equations: \(a-2 = 2√2\) and\(a-2 = −2√2\).

Step 4: Solve for a - Equation 1

For \(a-2 = 2√2\):\(a = 2 + 2√2\).

Step 5: Solve for a - Equation 2

For \(a-2 = -2√2\):\(a = 2 - 2√2\).

Key Concepts

Solving Equations

Solving equations is like solving puzzles. We have an unknown value, often represented by a letter like \( x \) or \( a \), and our job is to figure out what that value is. Equations can represent real-world problems and mathematical relationships. Here are some steps to solve equations:

• First, isolate the variable on one side of the equation. This allows us to find its value.
• Use mathematical operations like addition, subtraction, multiplication, and division to simplify.
• Always check your solutions by substituting them back into the original equation.

In the provided exercise, we started with the equation \( (a-2)^2 = 8 \). The goal was to find the value of \( a \) that makes this equation true. We used the even-root property to solve it, which leads us to our next section.

Square Root

The square root operation is the opposite of squaring a number. If you square 3, you get 9, and if you take the square root of 9, you get 3. Mathematically, this relationship is shown as: \( \text{if } x^2 = k, \text{ then } x = \boxed{\text{±}\text{sqrt}(k)} \).
In the exercise, we applied the square root to both sides of the equation \( (a-2)^2 = 8 \). Taking the square root of both sides, we get: \[ \text{sqrt} ((a-2)^2) = \text{±}\text{sqrt}(8) \]. This simplifies to: \[ |a-2| = 2\text{sqrt}(2) \]. This step is key because it makes the equation easier to solve. Keep in mind that taking the square root introduces both positive and negative solutions. These two potential values help us find all possible solutions to the original problem.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For example, \( |3| = 3 \) and \( |-3| = 3 \). Absolute value is represented by vertical bars around the number or expression, like |x|.
In solving the exercise, we reached \( |a-2| = 2\text{sqrt}(2) \). This implies two separate equations:

• \( a-2 = 2\text{sqrt}(2) \)
• \( a-2 = -2\text{sqrt}(2) \)

Solving these, we get the values of \( a \):

• For \( a-2 = 2\text{sqrt}(2) \), we add 2 to both sides, resulting in \( a = 2 + 2\text{sqrt}(2) \).
• For \( a-2 = -2\text{sqrt}(2) \), we add 2 to both sides, resulting in \( a = 2 - 2\text{sqrt}(2) \).

Understanding absolute value helps us to account for all possible solutions, ensuring no potential values are missed.