Thus, the coefficients are \( p = -3 \), \( q = 0 \), and \( r = 2 \).

["Understanding the Coefficients: A Deep Dive into the Quadratic Equation with ( p = -3 ), ( q = 0 ), and ( r = 2 )", "When analyzing quadratic equations of the form ( ax^2 + bx + c = 0 ), the coefficients ( a ), ( b ), and ( c ) play a crucial role in determining the behavior and solutions of the equation. In this article, we focus on the specific case where the coefficients are defined as ( p = -3 ), ( q = 0 ), and ( r = 2 ). While ( p ), ( q ), and ( r ) are sometimes labeled differently depending on system conventions, here they correspond to ( a ), ( b ), and ( c ) in standard form. This equivalence allows us to explore the quadratic equation’s properties with precision.", "First, let’s clarify the standard quadratic equation:\n[\nax^2 + bx + c = 0\n]\nSubstituting the given values, we rewrite the equation as:\n[\n-3x^2 + 0x + 2 = 0\n]\nSimplifying, this becomes:\n[\n-3x^2 + 2 = 0\n]", "### Rearranged into Standard Form", "The term with ( q = 0 ) (i.e., ( bx )) vanishes here, resulting in a simplified quadratic without a linear component. This has important implications for the graph, solutions, and method of solution.", "---", "### Solving the Equation", "We begin by isolating the ( x^2 ) term:\n[\n-3x^2 = -2\n]\nMultiplying both sides by (-1):\n[\n3x^2 = 2\n]\nThen dividing by 3:\n[\nx^2 = \frac{2}{3}\n]\nTaking the square root of both sides yields two real solutions:\n[\nx = \pm\sqrt{\frac{2}{3}} = \pm\frac{\sqrt{6}}{3}\n]\n(The denominator is rationalized for standard mathematical presentation.)", "---", "### Interpreting the Coefficients", "- ( p = -3 ): The negative leading coefficient indicates the parabola opens downward. This influences the function’s maximum value and the direction in which solutions lie.\n- ( q = 0 ): The absence of the linear term means the parabola is symmetric about the y-axis—its vertex lies on the vertical axis. This symmetry simplifies vertex analysis and root distribution.\n- ( r = 2 ): The constant term gives the y-intercept, where the graph crosses the y-axis at ( (0, 2) ).", "---", "### Graphical Insights", "The equation ( -3x^2 + 2 = 0 ) produces a downward-opening parabola intersecting the y-axis above the origin. With ( q = 0 ), there is no linear shift, so the axis of symmetry is ( x = 0 )—the y-axis. The roots at ( x = \pm\frac{\sqrt{6}}{3} \approx \pm 0.816 ) are symmetric about the origin, reflecting the quadratic’s behavior when the linear term is absent.", "---", "### Applications and Relevance", "Understanding quadratics with ( q = 0 ) is essential in many scientific and engineering contexts. For example, in physics, motion under idealized free-fall (without air resistance) may follow equations resembling ( -at^2 + v_0t + h_0 = 0 ), where velocity ( v_0 ) is zero—mirroring ( q = 0 ). Identifying such cases allows for faster modeling and solution.", "---", "### Conclusion", "The coefficients ( p = -3 ), ( q = 0 ), and ( r = 2 ) uniquely define a quadratic equation with a downward-opening parabola, no linear term, and y-intercept at 2. Solving ( -3x^2 + 2 = 0 ) reveals symmetric real roots, grounded in the symmetry and orientation imposed by the coefficient values. Recognizing these relationships enhances clarity in quadratic analysis—from solving to graphical interpretation—and underscores the power of coefficient-driven insight in algebra.", "---", "Further Reading:\n- Explore vertex form and completing the square for quadratics without linear terms.\n- Study symmetry properties of parabolas in coordinate geometry.\n- Investigate real-world modeling with simplified quadratic equations.", "---", "Keywords: quadratic equation solution, coefficients p q r, ( -3x^2 + 2 = 0 ), parabola symmetry, graph of quadratic with q=0, solving equations with no linear term."]