["Exploring the Quadratic Equation: Solving at ( a = 2 ), ( b = -8 ), ( c = 6 )", "Working with quadratic equations is fundamental in algebra, and understanding their roots, shape, and behavior helps in a wide range of applications—from physics to economics. In this article, we explore the specific quadratic expression defined by ( a = 2 ), ( b = -8 ), and ( c = 6 ), analyzing its solutions, graph, and importance in mathematical modeling.", "---", "### The Quadratic Equation Form", "The general form of a quadratic equation is:
\n[
\nax^2 + bx + c = 0
\n]
\nWith your values ( a = 2 ), ( b = -8 ), and ( c = 6 ), the equation becomes:
\n[
\n2x^2 - 8x + 6 = 0
\n]", "---", "### Simplifying the Equation", "Before solving, you can simplify the equation by dividing all terms by 2:
\n[
\nx^2 - 4x + 3 = 0
\n]", "This simplification makes it easier to factor or apply the quadratic formula.", "---", "### Factoring the Quadratic", "To factor ( x^2 - 4x + 3 ), look for two numbers that multiply to ( 3 ) and add to ( -4 ). These numbers are ( -1 ) and ( -3 ):
\n[
\nx^2 - 4x + 3 = (x - 1)(x - 3) = 0
\n]", "---", "### Solving for Roots", "Setting each factor equal to zero gives the solutions:
\n[
\nx - 1 = 0 \quad \Rightarrow \quad x = 1
\n]
\n[
\nx - 3 = 0 \quad \Rightarrow \quad x = 3
\n]", "So, the roots of the quadratic are ( x = 1 ) and ( x = 3 ). These x-intercepts define where the parabola crosses the x-axis.", "---", "### Graphing the Quadratic Function", "The graph of ( y = 2x^2 - 8x + 6 ) (original coefficients) is a parabola opening upwards because the coefficient of ( x^2 ) is positive (( a = 2 > 0 )).", "- Vertex: The x-coordinate of the vertex is given by ( x = -\frac{b}{2a} = -\frac{-8}{2 \cdot 2} = 2 ). Substituting ( x = 2 ) into the equation gives:
\n [
\n y = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2
\n ]
\n So, the vertex is at ( (2, -2) ).", "- Axis of Symmetry: The vertical line ( x = 2 ) bisects the parabola.", "- Y-intercept: When ( x = 0 ), ( y = 6 ), placing the y-intercept at ( (0, 6) ).", "---", "### Interpreting the Graph and Real-World Applications", "Because the parabola opens upward and has real, distinct roots, the quadratic models phenomena involving minima, such as profit maximization, motion under constant acceleration, or optimization problems.", "In real-world terms:
\n- The vertex at ( (2, -2) ) represents a minimum point in physical systems like rocket trajectories or cost functions.
\n- The roots ( x = 1 ) and ( x = 3 ) correspond to times or values where an event occurs—such as projectile hits or break-even points.", "---", "### Using the Quadratic Formula (Alternative Method)", "For completeness, applying the quadratic formula:
\n[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]
\nWith ( a = 2 ), ( b = -8 ), ( c = 6 ):
\n[
\nx = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 2 \cdot 6}}{2 \cdot 2} = \frac{8 \pm \sqrt{64 - 48}}{4} = \frac{8 \pm \sqrt{16}}{4} = \frac{8 \pm 4}{4}
\n]
\nThis yields:
\n[
\nx = \frac{12}{4} = 3 \quad \ ext{and} \quad x = \frac{4}{4} = 1
\n]
\nConfirming our factoring solution.", "---", "### Summary", "Working with ( a = 2 ), ( b = -8 ), ( c = 6 ) reveals important properties of a quadratic equation:
\n- The simplified form ( x^2 - 4x + 3 = 0 ) factors neatly to ( (x-1)(x-3) = 0 ).
\n- The roots ( x = 1 ) and ( x = 3 ) define the x-intercepts.
\n- The vertex at ( (2, -2) ) highlights the parabola’s minimum.
\n- Whether solved by factoring or the quadratic formula, this equation exemplifies core algebraic techniques and real-world applicability.", "Understanding these principles helps build a strong foundation in algebra—critical for more advanced mathematics and practical problem solving.", "---", "Keywords: quadratic equation, solve 2x² - 8x + 6 = 0, roots of quadratic, vertex form, graph parabola, factoring quadratic, quadratic formula, algebra tutorial", "---", "This comprehensive look at the quadratic with specific coefficients clarifies concepts essential for students, educators, and anyone interested in mastering algebra."]
