Problem 21 - Elementary Algebra

Question

Solve the following quadratic equations. \(\left(r-\frac{1}{2}\right)^{2}=\frac{3}{4}\)

Step-by-Step Solution

Answer

The solutions are \( r = \frac{1}{2} + \frac{\sqrt{3}}{2} \) and \( r = \frac{1}{2} - \frac{\sqrt{3}}{2} \).

Step 1: Expand the Equation

Given the equation \(\left(r-\frac{1}{2}\right)^{2}=\frac{3}{4}\), it's a quadratic equation in the form of \a^2 = b. To solve for r, eliminate the square by taking the square root of both sides.

Step 2: Take the Square Root of Both Sides

Taking the square root of both sides, we get: \(\sqrt{\left(r-\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{4}}\).Simplifying it, we have: \left|r-\frac{1}{2}\right|=\frac{\sqrt{3}}{2}.

Step 3: Set Up Two Equations

Since the absolute value of \(r - \frac{1}{2}\) can equal \(\frac{\sqrt{3}}{2}\) or \(-\frac{\sqrt{3}}{2}\), set up the two separate equations:1. \(r - \frac{1}{2} = \frac{\sqrt{3}}{2}\)2. \(r - \frac{1}{2} = -\frac{\sqrt{3}}{2}\).

Step 4: Solve for r

Solve each equation for r:1. \(r - \frac{1}{2} = \frac{\sqrt{3}}{2}\) \(r = \frac{1}{2} + \frac{\sqrt{3}}{2}\)2. \(r - \frac{1}{2} = -\frac{\sqrt{3}}{2}\) \(r = \frac{1}{2} - \frac{\sqrt{3}}{2}\).

Step 5: Write the Final Answer

The solutions are: \(\frac{1}{2} + \frac{\sqrt{3}}{2}, \frac{1}{2} - \frac{\sqrt{3}}{2}\).

Key Concepts

quadratic equations

Quadratic equations are expressions of the form \(ax^2 + bx + c = 0\). They involve terms with squared variables and can be solved using different methods such as factoring, completing the square, or the quadratic formula. In our given problem, the original equation is \((r - \frac{1}{2})^2 = \frac{3}{4}\). Here, we treat it as a basic quadratic form, \(a^2 = b\). Our goal is to isolate the variable, \(r\), and find its value.

square root

Taking the square root is a crucial step in solving quadratic equations, especially when the equation is already in the form \(a^2 = b\). By taking the square root of both sides of \((r - \frac{1}{2})^2 = \frac{3}{4}\), we obtain \(\sqrt{(r - \frac{1}{2})^2} = \sqrt{\frac{3}{4}}\). This allows us to simplify and remove the square exponent. Remember, taking the square root of both sides introduces both a positive and negative solution because both \(a\) and \(-a\) squared yield \(a^2\). For our example, this results in \(\left| r - \frac{1}{2} \right| = \frac{\sqrt{3}}{2}\).

absolute value

An absolute value equation involves expressions set within vertical bars, which measure the distance from zero regardless of direction. In our problem, \(\left| r - \frac{1}{2} \right| = \frac{\sqrt{3}}{2}\) implies two scenarios: \(r - \frac{1}{2} = \frac{\sqrt{3}}{2}\) and \(r - \frac{1}{2} = -\frac{\sqrt{3}}{2}\). Absolute value equations like these essentially break into two cases that must be resolved independently. This step is essential for ensuring all potential solutions are considered.

expand equation

Expanding equations helps simplify and solve them. Often, equations need to be rearranged to isolate the variable. For our initial problem, it's already simplified to \((r - \frac{1}{2})^2 = \frac{3}{4}\). Upon taking the square root and addressing the absolute value, we set up two separate linear equations: \(r - \frac{1}{2} = \frac{\sqrt{3}}{2}\) and \(r - \frac{1}{2} = -\frac{\sqrt{3}}{2}\). Solving these cases individually, we find \(r\) by further isolating it: \(r = \frac{1}{2} + \frac{\sqrt{3}}{2}\) and \(r = \frac{1}{2} - \frac{\sqrt{3}}{2}\). These steps lead us to the final solutions.