Divide by 2: \( 2x^2 + 35x - 102 = 0 \)

["# Solve ( 2x^2 + 35x - 102 = 0 ): Step-by-Step Division by 2 Method", "## Introduction\nSolving quadratic equations is a fundamental skill in algebra, and one common technique involves simplifying the equation by dividing through by the leading coefficient—this is known as the Divide by 2 (or more generally, Divide by the Leading Coefficient) method. In this article, we’ll walk through how to solve the quadratic equation:\n[ 2x^2 + 35x - 102 = 0 ]\nusing division by the leading coefficient to simplify solving.", "---", "## Why Divide by 2?\nNot all quadratics start with a coefficient of 1, which complicates factoring or using the quadratic formula. Dividing the entire equation by the leading coefficient—here, 2—produces a simpler form ideal for standard solution methods.", "---", "## Step 1: Write the Original Equation\nStart with:\n[ 2x^2 + 35x - 102 = 0 ]", "---", "## Step 2: Divide All Terms by 2\nDivide every term by 2:\n[\n\frac{2x^2}{2} + \frac{35x}{2} - \frac{102}{2} = \frac{0}{2}\n]\nSimplifying:\n[\nx^2 + \frac{35}{2}x - 51 = 0\n]", "---", "## Step 3: Solve the Simplified Quadratic Equation\nNow solve:\n[ x^2 + \frac{35}{2}x - 51 = 0 ]", "Since the coefficient of ( x ) is fractional, multiply the entire equation by 2 again to eliminate the fraction (this reverses the earlier divide but maintains equivalence):\n[\n2x^2 + 35x - 102 = 0\n]\nThis brings us back—but for clarity, we use the simplified form and apply the quadratic formula directly to:\n[\nx^2 + \frac{35}{2}x - 51 = 0\n]", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere\n( a = 1 ),\n( b = \frac{35}{2} ),\n( c = -51 ).", "---", "## Step 4: Plug Values into the Formula\nCalculate discriminant:\n[\nb^2 - 4ac = \left( \frac{35}{2} \right)^2 - 4(1)(-51) = \frac{1225}{4} + 204 = \frac{1225 + 816}{4} = \frac{2041}{4}\n]", "Now compute:\n[\nx = \frac{ -\frac{35}{2} \pm \sqrt{ \frac{2041}{4} } }{2}\n= \frac{ -\frac{35}{2} \pm \frac{\sqrt{2041}}{2} }{2}\n= \frac{ -35 \pm \sqrt{2041} }{4}\n]", "---", "## Final Solution\nThe two solutions to ( 2x^2 + 35x - 102 = 0 ) are:\n[\nx = \frac{ -35 + \sqrt{2041} }{4} \quad \ ext{and} \quad x = \frac{ -35 - \sqrt{2041} }{4}\n]", "---", "## Why This Works\nDividing by the leading coefficient simplifies the equation, making it easier to apply algebraic or formulaic methods without losing mathematical accuracy. While factoring may still be complex, dividing first helps prepare the expression clearly for calculation.", "---", "## Tips for Similar Equations\n- Always simplify by dividing through by the leading coefficient.\n- For non-integer coefficients, multiply through to eliminate fractions early.\n- Use the quadratic formula when factoring is difficult.\n- Check solutions by plugging back into the original equation.", "---", "## Summary\nSolving ( 2x^2 + 35x - 102 = 0 ) using divide-by-2 technique transforms a messy quadratic into a cleaner form, enabling efficient application of standard solving methods. Mastering this approach strengthens your algebraic toolkit for quadratic equations.", "---", "keyword: divide by 2, quadratic equation solution, ( 2x^2 + 35x - 102 = 0 ), solve quadratic by dividing, algebraic simplification, quadratic formula, step-by-step algebra", "---", "Ready to solve your next quadratic? Use divide by 2 to simplify and apply the quadratic formula confidently—efficiency meets accuracy!"]