["Understanding the Quadratic Equation with Coefficients (a = 2), (b = -3), (c = -2)", "When solving quadratic equations of the form (ax^2 + bx + c = 0), choosing specific values for the coefficients can make analysis easier and more educational. Consider the equation where (a = 2), (b = -3), and (c = -2). This setup forms the equation:", "[
\n2x^2 - 3x - 2 = 0
\n]", "This quadratic equation serves as a classic example for understanding key concepts like root finding, graph behavior, and factorization.", "### Solving the Equation", "To find the solutions, we can use the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plugging in the values:", "- (a = 2),
\n- (b = -3),
\n- (c = -2)", "Calculate the discriminant:", "[
\n\Delta = b^2 - 4ac = (-3)^2 - 4(2)(-2) = 9 + 16 = 25
\n]", "Since the discriminant is positive ((25 > 0)), there are two distinct real roots. Applying the formula:", "[
\nx = \frac{-(-3) \pm \sqrt{25}}{2 \ imes 2} = \frac{3 \pm 5}{4}
\n]", "So, the two solutions are:", "- (x = \frac{3 + 5}{4} = \frac{8}{4} = 2)
\n- (x = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2})", "### Factoring the Quadratic Expression", "Thanks to the rational roots, we can factor the quadratic expression:", "[
\n2x^2 - 3x - 2 = (2x + 1)(x - 2)
\n]", "Expanding this confirms:", "[
\n(2x + 1)(x - 2) = 2x^2 - 4x + x - 2 = 2x^2 - 3x - 2
\n]", "Factoring allows us to identify roots by setting each factor to zero:", "- (2x + 1 = 0 \Rightarrow x = -\frac{1}{2})
\n- (x - 2 = 0 \Rightarrow x = 2)", "### Graphical Representation", "The corresponding quadratic function is (f(x) = 2x^2 - 3x - 2). Its graph is a parabola opening upwards because the coefficient (a = 2 > 0). The vertex lies between the roots, and the axis of symmetry is:", "[
\nx = -\frac{b}{2a} = \frac{3}{4}
\n]", "Evaluating (f\left(\frac{3}{4}\right)) gives the minimum point, reinforcing how the question’s coefficients define a parabola crossing the x-axis at (x = -\frac{1}{2}) and (x = 2).", "### Practical Applications", "Such equations model real-world phenomena including projectile motion, profit functions, and physics problems. With (a = 2), (b = -3), (c = -2), students and professionals gain clarity on behavior of parabolas, solving for unknowns, and interpreting impact values.", "### Conclusion", "Working with specific coefficients like (a = 2), (b = -3), (c = -2) simplifies quadratic analysis while illustrating core algebraic principles—solving via the quadratic formula, factoring, graphing, and understanding real-world applications. Whether for teaching or problem-solving, this structured example provides a solid foundation in quadratic equations.", "---", "Keywords: quadratic equation 2x²−3x−2=0, solving quadratic formula, factoring quadratics, discriminant, roots of quadratic, projectile motion example, real numbers, a=2, b=-3, c=-2."]
