The solutions are \( x = 3 \) and \( x = -0.5 \).

["Understanding the Solutions: ( x = 3 ) and ( x = -0.5 )", "When solving equations, particularly linear or quadratic equations, finding the precise solutions is key to understanding the behavior of the function or relationship described. In this case, the solutions ( x = 3 ) and ( x = -0.5 ) represent the points where a function intersects the x-axis—also known as the x-intercepts. This article explores what these solutions mean, how they are derived, and why they matter in algebra and real-world applications.", "---", "### What Do ( x = 3 ) and ( x = -0.5 ) Represent?", "The values ( x = 3 ) and ( x = -0.5 ) are the roots or solutions of the equation:\n[\nx = 3 \quad \ ext{or} \quad x = -0.5\n]\nI.e., they satisfy an equation such as:\n[\nx - 3 = 0 \quad \ ext{or} \quad 2x + 0.5 = 0\n]", "These points are critical because they indicate where the graph of the function crosses or touches the x-axis—fundamental concepts in graphing and analysis of functions.", "---", "### Step-by-Step: How to Find Solutions ( x = 3 ) and ( x = -0.5 )", "#### Case 1: Linear Equation\nFor example, if the equation is:\n[\n2x - 9 = 0\n]\nSolving:\n[\n2x = 9 \quad \Rightarrow \quad x = \frac{9}{2} = 4.5\n]\nThis doesn’t match the given solutions. But if we solve:\n[\nx + 0.5x - 3 = 0 \quad \Rightarrow \quad 1.5x = 3 \quad \Rightarrow \quad x = 2\n]\nStill not matching.", "Actually, if ( x = 3 ) and ( x = -0.5 ) are the two solutions, a quadratic equation of the form:\n[\n(x - 3)(x + 0.5) = 0\n]\nexpanding gives:\n[\nx^2 + 0.5x - 3x - 1.5 = x^2 - 2.5x - 1.5 = 0\n]\nThe roots are clearly ( x = 3 ) and ( x = -0.5 ). So the equation whose solutions are ( x = 3 ) and ( x = -0.5 ) is:\n[\nx^2 - 2.5x - 1.5 = 0\n]", "---", "### Why Are These Solutions Important?", "1. Graphing Insight\n Knowing ( x = 3 ) and ( x = -0.5 ) as roots allows you to plot the corresponding x-intercepts. This visual representation is essential in analyzing trends, maxima, minima, and overall function behavior.", "2. Problem Solving\n In real-world scenarios—whether modeling projectile motion, economics, or population growth—these roots often represent critical thresholds or breaking points where outcomes change dramatically.", "3. Verification\n Plugging in ( x = 3 ) and ( x = -0.5 ) into the original equation confirms correctness. For instance, if used in a real equation, both values satisfy it exactly.", "---", "### Real-World Applications", "- Engineering: Finding break-even points in cost-revenue models.\n- Physics: Determining time points where displacement is zero in motion equations.\n- Economics: Identifying price points where profit is zero.", "---", "### Conclusion", "The solutions ( x = 3 ) and ( x = -0.5 ) are more than just numbers—they are pivotal points where functions intersect the x-axis and reveal critical information about the system being modeled. Whether through direct solving or constructing equations from roots, understanding these solutions strengthens problem-solving skills and deepens mathematical insight.", "If you're working with these solutions, use them to plot graphs, verify equations, or explore real-world implications. Mastering such roots is fundamental to advancing in algebra and applied mathematics.", "---", "Keywords:\nx = 3 solutions, x = -0.5, roots of equations, solving linear equations, quadratic roots, function intercepts, algebra solutions, real-world applications, graphing equations", "Meta Description:\nUnderstand the significance of the solutions ( x = 3 ) and ( x = -0.5 ), how they are derived, and their importance in algebra and real-world applications. Explore step-by-step analysis and practical relevance."]