Since \( a = 2 \), multiply the entire equation by 2: \( 2x^2 + 4x - 30 = 0 \).

["SEO-Optimized Article: Simplifying the Quadratic Equation ( 2x^2 + 4x - 30 = 0 ) When ( a = 2 )", "When solving quadratic equations, manipulation and simplification play a key role in making complex expressions easier to work with. In this guide, we focus on a specific case where the leading coefficient ( a = 2 ), transforming the quadratic equation ( 2x^2 + 4x - 30 = 0 ) into a cleaner form: by multiplying the entire equation by 2. This process not only streamlines solving but also enhances clarity for students, educators, and algebraic learners.", "### Why Multiply by ( a )?\nIn standard quadratic form, an equation takes the shape:\n[ ax^2 + bx + c = 0 ]\nHere, ( a ) determines the parabola’s width and direction. If ( a = 0 ), it ceases to be quadratic. However, multiplying both sides of a non-zero quadratic equation by ( a ) maintains equality and transforms the equation without altering its solution set.", "For the equation ( 2x^2 + 4x - 30 = 0 ), we are told ( a = 2 ), so clearly ( a <br/>\neq 0 ), allowing safe multiplication.", "### Step-by-Step Multiplication\nStarting with:\n[\n2x^2 + 4x - 30 = 0\n]\nMultiply every term by 2:\n[\n2 \cdot (2x^2) + 2 \cdot (4x) + 2 \cdot (-30) = 2 \cdot 0\n]\n[\n4x^2 + 8x - 60 = 0\n]", "While this results in a new quadratic expression, the transformation offers significant advantages:", "- Easier root-finding: The wider coefficient range simplifies applying the quadratic formula.\n- Cleaner intermediate form: The simplified equation may better align with factoring or completing the square techniques.\n- Consistency in coefficient relationships: The ratio ( a : b : c = 4 : 8 : -60 ) preserves proportional relationships for substitution methods.", "### Benefits in Problem Solving\nMultiplying by ( a = 2 ) prepares the equation for advanced solving strategies:", "- Quadratic Formula: Plugging into ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), with doubled coefficients avoids fractional inconsistencies.\n[\na = 4,\ b = 8,\ c = -60 \Rightarrow x = \frac{-8 \pm \sqrt{8^2 - 4(4)(-60)}}{2(4)} = \frac{-8 \pm \sqrt{64 + 960}}{8} = \frac{-8 \pm \sqrt{1024}}{8} = \frac{-8 \pm 32}{8}\n]\nSolutions:\n[\nx = \frac{24}{8} = 3 \quad \ ext{and} \quad x = \frac{-40}{8} = -5\n]", "- Factoring & Finite Difference Methods: A simplified equation often reveals factors more quickly. Here, the equation ( 4x^2 + 8x - 60 = 0 ) factors as ( 4(x^2 + 2x - 15) = 0 ), leading swiftly to ( x^2 + 2x - 15 = 0 ) and factoring to ( (x + 5)(x - 3) = 0 ).", "### Educational Takeaway\nMultiplying a quadratic equation by its leading coefficient when ( a <br/>\ne 0 ) is a strategic algebraic tool. It does not change the roots but simplifies downstream computation—ideal for students mastering quadratic solutions and instructors designing clear problem sets.", "Keywords: quadratic equation, solving ( 2x^2 + 4x - 30 = 0 ), multiply by ( a ), simplifying quadratics, quadratic formula, algebraic manipulation, equation transformation, factoring methods.", "---", "By applying this essential technique, learners gain clearer access to quadratic results—empowering accurate solutions and deeper conceptual understanding."]