\[ (2x - 1)(x - 1) = 0. \] - United Radiology

["# Solving the Equation: (2x - 1)(x - 1) = 0", "Understanding how to solve quadratic equations is essential for mastering algebra. One of the most straightforward methods for solving equations like [(2x - 1)(x - 1) = 0] is by applying the Zero Product Property. This principle makes it easy to find all possible values of (x) that make the expression equal to zero.", "## What Does (2x - 1)(x - 1) = 0 Mean?", "The equation [(2x - 1)(x - 1) = 0] states that the product of two factors, (2x - 1) and (x - 1), equals zero. According to the Zero Product Property, if a product of factors is zero, then at least one of the factors must be zero.", "## Step-by-Step Solution", "### Step 1: Set Each Factor Equal to Zero
\nSet each factor separately equal to zero:
\n1. (2x - 1 = 0)
\n2. (x - 1 = 0)", "### Step 2: Solve the First Equation
\nFrom (2x - 1 = 0), add 1 to both sides:
\n[ 2x = 1 ]
\nThen divide by 2:
\n[ x = \frac{1}{2} ]", "### Step 3: Solve the Second Equation
\nFrom (x - 1 = 0), simply add 1:
\n[ x = 1 ]", "## Final Solution", "The solutions to the equation [(2x - 1)(x - 1) = 0] are:
\n[ x = \frac{1}{2} \quad \ ext{and} \quad x = 1 ]", "## Why These Solutions Matter", "Each solution represents a critical point where the expression changes sign. Graphically, these values are the roots of the corresponding quadratic function (f(x) = (2x - 1)(x - 1)), meaning the graph crosses or touches the x-axis at (x = \frac{1}{2}) and (x = 1). Understanding these roots helps in analyzing functions, optimizing values, and solving real-world problems.", "## How to Graph the Equation", "When graphing [(2x - 1)(x - 1)], plot the points where (x = \frac{1}{2}) and (x = 1). Since this is a quadratic expression, the graph forms a parabola opening upwards (as the leading coefficient is positive). The roots at (x = \frac{1}{2}) and (x = 1) are where the parabola intersects the x-axis.", "## Conclusion", "Solving [(2x - 1)(x - 1) = 0] is simple using the Zero Product Property. By solving each factor, we find the solutions (x = \frac{1}{2}) and (x = 1), which are key to understanding the behavior of the quadratic function. Whether you're studying algebra for school or tackling math in real life, mastering such equations builds a strong foundation in problem solving.", "---", "Keywords: solve (2x - 1)(x - 1) = 0, quadratic equation solutions, zero product property, algebra tutorial, solving equations step-by-step, roots of quadratic equation, graphing quadratics."]