Problem 24 - Algebra for College Students

Question

Use the even-root property to solve each equation. $$\left(w+\frac{2}{3}\right)^{2}=\frac{5}{9}$$

Step-by-Step Solution

Answer

The solutions are \( w = \frac{\sqrt{5} - 2}{3} \) and \( w = \frac{-\sqrt{5} - 2}{3} \).

Step 1: Understand the Even-Root Property

The even-root property states that if \( x^2 = k \), then \( x = \pm\sqrt{k} \). This means the solution can be either the positive or negative square root of k.

Step 2: Apply the Even-Root Property

Given the equation \( \left(w + \frac{2}{3}\right)^2 = \frac{5}{9} \), apply the even-root property: \( w + \frac{2}{3} = \pm \sqrt{\frac{5}{9}} \).

Step 3: Simplify the Square Root

Simplify the right-hand side: \( \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \). So the equation becomes \( w + \frac{2}{3} = \pm \frac{\sqrt{5}}{3} \).

Step 4: Solve for w

Isolate \( w \) by subtracting \( \frac{2}{3} \) from both sides of the equation. Thus, \( w = \pm \frac{\sqrt{5}}{3} - \frac{2}{3} \).

Step 5: Write the Final Solution

The values of \( w \) are \( \frac{\sqrt{5}}{3} - \frac{2}{3} \) and \(-(\frac{\sqrt{5}}{3}) - \frac{2}{3} \). Therefore, the solution is \( w = \frac{\sqrt{5} - 2}{3} \) and \( w = \frac{-\sqrt{5} - 2}{3} \).

Key Concepts

solving equations

Solving equations is a fundamental skill in algebra that involves finding the values of variables that make an equation true. The core steps generally involve:

• Understanding the equation
• Applying mathematical properties or rules
• Isolating the variable
• Verifying the solution

In our example, the key was to use the even-root property, which helps to simplify equations involving squared terms.
By following systematic steps, you can handle even more complex equations effectively. Practicing these steps helps in developing a strong foundation in algebra.

square roots

Square roots are numbers that, when multiplied by themselves, give the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Taking a square root is the opposite operation of squaring a number.
When you have an equation like \((w + \frac{2}{3})^{2} = \frac{5}{9}\), the goal is to remove the square by applying the square root. But remember, \(\sqrt{k}\) can be both positive and negative. This is crucial as it gives you two possible solutions.
Simplifying the square root term \( \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \) helps in making the equation easier to solve. Always break down complex terms into simpler forms whenever possible.

isolation of variable

Isolating the variable means getting the variable alone on one side of the equation. This often involves:

• Adding or subtracting terms on both sides
• Multiplying or dividing terms
• Simplifying fractions or radicals

In the equation \((w + \frac{2}{3})^{2} = \frac{5}{9}\), isolating the variable involves taking the square root of both sides and then subtracting \(\frac{2}{3}\) from both sides.
This step-by-step approach makes the equation simpler and helps find the precise values for the variable.