Problem 23 - Algebra for College Students

Question

Use the even-root property to solve each equation. $$\left(w-\frac{3}{2}\right)^{2}=\frac{7}{4}$$

Step-by-Step Solution

Answer

w = \frac{3}{2} + \frac{1}{2}\root{4}{7}, w = \frac{3}{2} - \frac{1}{2}\root{4}{7}.

Step 1: Isolate the square term

The equation is already in the form \(\big(w - \frac{3}{2}\big)^2 = \frac{7}{4}\). This means we can proceed to the next step directly.

Step 2: Apply the even-root property

Taking the square root of both sides, we have \(\big|w - \frac{3}{2}\big| = \frac{\root{4}{7}}{2}\).

Step 3: Solve for the variable

This absolute value equation can be split into two linear equations: \(\big(w - \frac{3}{2}\big) = \frac{1}{2}\root{4}{7}\) and \(\big(w - \frac{3}{2}\big) = -\frac{1}{2}\root{4}{7}\).

Step 4: Solve the first linear equation

For \(\big(w - \frac{3}{2}\big) = \frac{1}{2}\root{4}{7}\), we add \(\frac{3}{2}\) to both sides to get: \w = \frac{3}{2} + \frac{1}{2}\root{4}{7}\.

Step 5: Solve the second linear equation

For \(\big(w - \frac{3}{2}\big) = -\frac{1}{2}\root{4}{7}\), we add \(\frac{3}{2}\) to both sides to get: \w = \frac{3}{2} - \frac{1}{2}\root{4}{7}\.

Step 6: State the solutions

The solutions to the equation are \(\frac{3}{2} + \frac{1}{2}\root{4}{7}\) and \(\frac{3}{2} - \frac{1}{2}\root{4}{7}\).

Key Concepts

solving equations

Solving equations is a fundamental skill in algebra and higher mathematics. An equation is a statement that two expressions are equal, and solving an equation involves finding the value(s) of the variable(s) that make the equation true. To solve an equation:

• First, isolate the variable by performing operations to simplify the expression.
• Next, apply inverse operations systematically to solve for the variable.
• Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

In the given exercise, isolating the squared term is essential before performing any operations. Once the term is isolated, applying properties like the even-root property allows further simplification. This brings us to solving an absolute value equation, which involves two potential solutions as absolute value expressions can have positive and negative interpretations.

absolute value equations

Absolute value equations contain expressions within absolute value signs, which impose additional considerations during solution steps. The absolute value of a number measures its distance from zero on the number line, ignoring any negative sign. Therefore, when solving equations involving absolute values:

• The expression inside the absolute value can be equal to either the positive or the negative value of the number on the other side of the equation.
• This creates two scenarios to solve, resulting in two linear equations.

In the exercise, \(|w - \frac{3}{2}| = \frac{1}{2}\sqrt{7}\) results in two separate equations: one for the positive case and one for the negative case. These form:

\(w - \frac{3}{2} = \frac{1}{2}\sqrt{7}\)

and

\(w - \frac{3}{2} = -\frac{1}{2}\sqrt{7}\).

Each of these linear equations needs to be solved individually to find both potential solutions for the variable.

square root

The square root is a fundamental mathematical operation that finds the value that, when multiplied by itself, results in the original number. Specifically, if \(x^{2} = y\), then \(x = \pm\sqrt{y}\). Key points to remember include:

• Taking the square root of both sides of an equation can help simplify and solve it, but remember to consider both positive and negative roots.
• The notation \(\sqrt{y}\) represents the principal (positive) square root.
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