Divide by 2: \( x^2 + x - 42 = 0 \)

["# Solving the Quadratic Equation: Divide by 2 and Solve ( x^2 + x - 42 = 0 )", "When faced with a quadratic equation like ( x^2 + x - 42 = 0 ), simplifying or dividing the equation by a constant can make solving it much easier. In this article, we’ll explore how dividing the entire equation by 2 changes its appearance—and verify whether this step truly simplifies solving for ( x ). We’ll walk through the standard methods for solving quadratics, clarify the impact of dividing by 2, and provide a clear, step-by-step solution.", "---", "## Why Divide 2? Understanding the Context", "The equation ( x^2 + x - 42 = 0 ) is already in standard quadratic form ( ax^2 + bx + c = 0 ) with ( a = 1 ), ( b = 1 ), ( c = -42 ). Dividing the entire equation by 2 results in:", "[\n\frac{1}{2}x^2 + \frac{1}{2}x - 21 = 0\n]", "While this transforms the coefficients into fractions, it does not generally make solving easier. In fact, clearing denominators by multiplying through by the denominator often simplifies calculations — but dividing introduces fractions that complicate factoring or quadratic formula applications.", "Key point: Dividing by 2 doesn’t simplify solving — in most cases, multiplying through by 2 (to eliminate fractions) is preferred. However, understanding both forms deepens insight into quadratic behavior.", "---", "## Step 1: Use the Quadratic Formula", "The standard quadratic formula solves ( ax^2 + bx + c = 0 ) with:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 1 ), ( b = 1 ), ( c = -42 ):", "[\nx = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2}\n]", "Since ( \sqrt{169} = 13 ):", "[\nx = \frac{-1 \pm 13}{2}\n]", "---", "## Step 2: Calculate the Two Solutions", "[\nx = \frac{-1 + 13}{2} = \frac{12}{2} = 6\n]\n[\nx = \frac{-1 - 13}{2} = \frac{-14}{2} = -7\n]", "---", "## Step 3: Verification (Optional but Recommended)", "Plug the solutions back into the original equation:", "- For ( x = 6 ):\n ( (6)^2 + 6 - 42 = 36 + 6 - 42 = 0 ) ✅", "- For ( x = -7 ):\n ( (-7)^2 + (-7) - 42 = 49 - 7 - 42 = 0 ) ✅", "Both roots satisfy the equation.", "---", "## Bonus: Alternative – Factoring with Whole Coefficients", "Noticing ( x^2 + x - 42 = 0 ), we seek two numbers multiplying to -42 and adding to +1. These numbers are +7 and -6:", "[\nx^2 + x - 42 = (x + 7)(x - 6) = 0\n]", "Set each factor to zero:", "[\nx + 7 = 0 \Rightarrow x = -7\n\quad \ ext{or} \quad\nx - 6 = 0 \Rightarrow x = 6\n]", "Factoring is simpler when integers are involved—and as shown, ( (x + 7)(x - 6) = 0 ) avoids fractions entirely.", "---", "## Conclusion: When to Divide by 2?", "Dividing ( x^2 + x - 42 = 0 ) by 2 does not improve simplification—fractional coefficients often complicate calculations. Instead, factoring directly or applying the quadratic formula is preferable. However, dividing by 2 temporarily transforms the equation to:", "[\n\frac{1}{2}x^2 + \frac{1}{2}x - 21 = 0\n]", "while preserving equivalence. While not optimal for hand-solving, multiplying through by 2 (i.e., multiplying all terms by 2) to clear fractions—resulting in ( x^2 + x - 42 = 0 )—is a standard and effective step for clarity before applying the quadratic formula.", "---", "## Summary Tools for This Problem:", "- Quadratic Formula: Essential for any ( ax^2 + bx + c = 0 ), especially when factoring is difficult.\n- Factoring by Inspection: Efficient when coefficients allow quick identification of factor pairs.\n- Multiplying to Clear Denominators: Simplifies equation form—better to multiply by 2 than divide.\n- Verification Step: Always confirm roots satisfy the original equation.", "---", "## Takeaways", "- Dividing a quadratic by 2 changes the coefficient values but preserves the solution set.\n- For ( x^2 + x - 42 = 0 ), factoring delivers the most straightforward solutions: ( x = -7 ) and ( x = 6 ).\n- Master both dividing and multiplying strategies to approach quadratic equations flexibly.\n- Always verify roots to ensure accuracy.", "Effortlessly solving ( x^2 + x - 42 = 0 ) means recognizing when denominators complicate rather than simplify—and choosing the right algebraic tool for the task.", "---", "Keywords: divide by 2, quadratic equation ( x^2 + x - 42 = 0 ), solve quadratic, quadratic formula, factoring, algebraic solutions."]