Here, \( a = 2 \), \( b = -8 \), and \( c = 6 \).

["# Exploring the Quadratic Equation with Coefficients ( a = 2 ), ( b = -8 ), and ( c = 6 )", "When studying quadratic equations, understanding the roles of the coefficients ( a ), ( b ), and ( c ) is essential for solving and graphing these fundamental expressions. In this article, we’ll explore the specific quadratic function defined by the coefficients ( a = 2 ), ( b = -8 ), and ( c = 6 ). This equation serves as a prime example for learning how quadratic behavior is shaped by each parameter and how it influences key features such as roots, vertex, and axis of symmetry.", "## The Standard Form of a Quadratic Equation", "A quadratic equation is generally written as:", "[\nf(x) = ax^2 + bx + c\n]", "Given ( a = 2 ), ( b = -8 ), and ( c = 6 ), we substitute these values into the standard form:", "[\nf(x) = 2x^2 - 8x + 6\n]", "This equation represents a parabola that opens upwards since the coefficient ( a = 2 > 0 ).", "## Key Features of the Quadratic Function", "### 1. The Vertex and Axis of Symmetry", "The vertex is the turning point of the parabola and gives the minimum (or maximum) value of the function. The x-coordinate of the vertex is calculated using the formula:", "[\nx = -\frac{b}{2a}\n]", "Substituting ( b = -8 ) and ( a = 2 ):", "[\nx = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2\n]", "To find the y-coordinate (the function value at the vertex), substitute ( x = 2 ) back into the equation:", "[\nf(2) = 2(2)^2 - 8(2) + 6 = 2 \cdot 4 - 16 + 6 = 8 - 16 + 6 = -2\n]", "Thus, the vertex is at ( (2, -2) ), confirming the axis of symmetry is the vertical line:", "[\nx = 2\n]", "### 2. Finding the Roots (Zeros)", "The roots of the quadratic equation are the ( x )-values where ( f(x) = 0 ). Using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Compute the discriminant first:", "[\n\Delta = (-8)^2 - 4(2)(6) = 64 - 48 = 16\n]", "Since the discriminant is positive, there are two real roots. Now calculate:", "[\nx = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 2} = \frac{8 \pm 4}{4}\n]", "So the two roots are:", "[\nx_1 = \frac{8 + 4}{4} = \frac{12}{4} = 3\n]", "[\nx_2 = \frac{8 - 4}{4} = \frac{4}{4} = 1\n]", "The equation factors as:", "[\nf(x) = 2(x - 1)(x - 3)\n]", "This confirms the roots ( x = 1 ) and ( x = 3 ).", "### 3. Graph Behavior and Transformations", "With the vertex at ( (2, -2) ) and roots at ( x = 1 ) and ( x = 3 ), the parabola crosses the x-axis between these two points. Since ( a = 2 ), the parabola is stretched vertically, making it narrower than the standard ( x^2 ) graph.", "The function value at ( x = 0 ) shows the y-intercept:", "[\nf(0) = 2(0)^2 - 8(0) + 6 = 6\n]", "So the graph passes through the point ( (0, 6) ).", "---", "## Summary Table of Key Values", "| Parameter | Value | Explanation |\n|-----------|-------|-------------|\n| Coefficient ( a ) | 2 | Opens the parabola upwards; controls vertical stretch |\n| Coefficient ( b ) | -8 | Determines position of the vertex and axis of symmetry |\n| Coefficient ( c ) | 6 | Y-intercept of the function at ( (0, 6) ) |\n| Vertex | ( (2, -2) ) | Minimum point due to positive ( a ); axis of symmetry ( x = 2 ) |\n| Roots | ( x = 1 ) and ( x = 3 ) | Points where parabola crosses x-axis |\n| Discriminant | 16 | Indicates two real distinct roots |", "---", "## Practical Use and Further Exploration", "Understanding this quadratic helps students and educators alike by illustrating how coefficients shape real-world graphs such as projectile trajectories, profit margins, or maximum height problems. Customizing ( a ), ( b ), and ( c ) allows exploration of different parabolic patterns, reinforcing algebraic and graphical interpretation.", "For further learning, try replacing ( a = 2 ), ( b = -8 ), ( c = 6 ) with other values or graph these points to visualize transformations and the effects of coefficient changes.", "---", "## Conclusion", "With ( a = 2 ), ( b = -8 ), and ( c = 6 ), the quadratic equation ( 2x^2 - 8x + 6 ) defines a parabola opening upwards with a vertex at ( (2, -2) ), roots at ( x = 1 ) and ( x = 3 ), and a meaningful y-intercept of ( (0, 6) ). Mastering such equations strengthens core algebra skills and opens doors to applied mathematics across science and engineering fields.", "---", "Keywords: quadratic equation, ( a=2 ), ( b=-8 ), ( c=6 ), vertex, roots, discriminant, quadratic graph, vertex form, algebra, coordinate geometry.\nMeta Description: Dive into the quadratic equation with ( a = 2 ), ( b = -8 ), ( c = 6 ). Learn about vertex, roots, and graph behavior step-by-step.\nTarget Audience: High school students, educators, and learners interested in algebra and quadratic graphs."]