x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}

["# Solving Quadratic Equations: A Step-by-Step Guide to Completing the Equation \nSolving ( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} )", "Quadratic equations are fundamental in algebra, appearing in fields ranging from engineering to economics. Solving them using the quadratic formula offers clarity and precision. Today, we’ll break down the specific quadratic equation:", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}\n]", "Understanding this equation not only helps solve for ( x ) but also deepens your grasp of quadratic functions, the discriminant, and algebraic reasoning. Let’s go through the step-by-step process.", "---", "### Understanding the Quadratic Formula", "The standard form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides the roots (solutions) of any quadratic equation. Here,\n- ( a ), ( b ), and ( c ) are coefficients from the quadratic equation.\n- The discriminant ( D = b^2 - 4ac ) determines the nature of the roots (real, repeated, or complex).", "---", "### Step 1: Identify Coefficients in the Given Equation", "From the provided equation:", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}\n]", "Compare it to ( ax^2 + bx + c = 0 ):\n- ( a = 2 )\n- ( b = -4 )\n- ( c = -6 )", "Note: The numerator uses (-(-4)), meaning ( -b = -(-4) = +4 ).", "---", "### Step 2: Substitute Coefficients into the Formula", "Now replace ( a ), ( b ), and ( c ) into the quadratic formula:", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}\n]", "Simplify each component:\n- ( -b = -(-4) = +4 )\n- Discriminant:\n [\n (-4)^2 = 16, \quad 4 \cdot 2 \cdot (-6) = -48, \quad \ ext{so } D = 16 - (-48) = 16 + 48 = 64\n ]\n- Denominator: ( 2a = 2 \cdot 2 = 4 )", "Thus, the equation becomes:", "[\nx = \frac{4 \pm \sqrt{64}}{4}\n]", "---", "### Step 3: Simplify the Square Root and Final Expression", "Calculate ( \sqrt{64} = 8 ), so:", "[\nx = \frac{4 \pm 8}{4}\n]", "Now compute both possible values:\n- ( x = \frac{4 + 8}{4} = \frac{12}{4} = 3 )\n- ( x = \frac{4 - 8}{4} = \frac{-4}{4} = -1 )", "---", "### Step 4: Interpret the Solutions", "The solutions to the equation are:\n[\nx = 3 \quad \ ext{and} \quad x = -1\n]", "These values represent where the quadratic function ( f(x) = 2x^2 - 4x - 6 ) crosses the x-axis (its roots). Understanding these points is essential in graphing and analyzing quadratic behavior.", "---", "### Why This Equation Matters", "This problem illustrates how the quadratic formula systematically solves any standard quadratic equation, regardless of factorability. Recognizing the discriminant’s role helps predict root types:\n- Positive discriminant → two distinct real roots (as here)\n- Zero → one repeated real root\n- Negative → complex conjugate roots", "Mastering this process empowers students and professionals to confidently solve quadratic problems in mathematics, physics, engineering, and beyond.", "---", "### Key Takeaways", "- Use the quadratic formula for any ( ax^2 + bx + c = 0 ).\n- Always carefully identify coefficients ( a ), ( b ), and ( c ).\n- Simplify step-by-step to avoid errors.\n- The discriminant reveals the nature of solutions.\n- Solved roots are critical for graphing and real-world applications.", "---", "### Final Notes", "Mastering quadratic equations like solving ( x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} ) equips you with a powerful algebraic tool. Practice with different coefficients to strengthen fluency—each problem sharpens logical reasoning and technical skill.", "---", "Learn more about quadratic equations and expand your algebra toolkit today!\nSample problems, formulas, and applications await—keep solving!", "---", "Keywords for SEO: quadratic formula, solve quadratic equation, discriminant, algebra 2, quadratic roots, completing the square, quadratic equations explained, x equals format, mathematics tutorial, solving quadratics step-by-step."]