x = \frac{-35 \pm \sqrt{1225 + 1400}}{4} = \frac{-35 \pm \sqrt{2625}}{4}

["Understanding the Simplified Quadratic Solution: ( x = \frac{-35 \pm \sqrt{2625}}{4} )", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), one of the most effective tools is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In this article, we explore a specific instance of this formula where:", "[\nx = \frac{-35 \pm \sqrt{1225 + 1400}}{4}\n]", "After evaluating the expression under the square root, we find that:", "[\n1225 + 1400 = 2625\n]", "Thus, the quadratic equation simplifies elegantly to:", "[\nx = \frac{-35 \pm \sqrt{2625}}{4}\n]", "---", "### Breaking Down the Expression: ( \sqrt{2625} )", "At first glance, ( \sqrt{2625} ) may seem challenging, but simplifying the square root reveals clearer insight.", "Start by factoring 2625:", "[\n2625 = 25 \ imes 105\n]", "Now simplify:", "[\n\sqrt{2625} = \sqrt{25 \ imes 105} = \sqrt{25} \ imes \sqrt{105} = 5\sqrt{105}\n]", "So, the solution becomes:", "[\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "This form highlights the irrational component and prepares us for practical computation or further algebraic manipulation.", "---", "### Why This Form Matters: Simplification and Applications", "Expressing the roots in simplified radical form offers several advantages:", "- Exact values: Avoids decimal approximations, preserving precision.\n- Easier computation: Useful in symbolic math and engineering where exact solutions are preferred.\n- Graphical and analytical analysis: Helps in determining the vertex, axis of symmetry (( x = \frac{-35}{4} )), and whether the roots are real and rational, irrational, or complex.", "Since ( \sqrt{105} ) cannot be simplified further (105 factors into 3 × 5 × 7, with no perfect square factors), ( \frac{-35 \pm 5\sqrt{105}}{4} ) represents the most reduced exact form.", "---", "### How to Use This Solution in Real Problems", "Suppose you’re solving:", "[\nx^2 + 35x + 2625 = 0\n]", "Here, ( a = 1 ), ( b = 35 ), ( c = 2625 ), confirming the original expression. Using the simplified quadratic formula:", "[\nx = \frac{-35 \pm \sqrt{2625}}{4} = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "This gives two precise solutions:", "[\nx_1 = \frac{-35 + 5\sqrt{105}}{4}, \quad x_2 = \frac{-35 - 5\sqrt{105}}{4}\n]", "These can be used in quadratic modeling, optimization problems, physics equations, or any context requiring exact solutions.", "---", "### Conclusion", "The equation:", "[\nx = \frac{-35 \pm \sqrt{1225 + 1400}}{4} = \frac{-35 \pm \sqrt{2625}}{4}\n]", "demonstrates the power of simplifying radicals to enhance clarity and utility. Recognizing and reducing square roots like ( \sqrt{2625} = 5\sqrt{105} ) transforms abstract expressions into actionable, exact solutions. Whether in algebra, calculus, or applied sciences, mastering such forms empowers deeper mathematical understanding and problem-solving accuracy.", "Keywords: quadratic formula, ( x = \frac{-35 \pm \sqrt{2625}}{4} ), simplify radicals, solving quadratics, exact solutions, ( \sqrt{105} ), exact math, algebraic simplification", "---", "Want to calculate these roots numerically?", "- ( \sqrt{2625} \approx 51.23 )\n- Then ( x \approx \frac{-35 \pm 51.23}{4} )\n- ( x_1 \approx \frac{16.23}{4} \approx 4.06 )\n- ( x_2 \approx \frac{-86.23}{4} \approx -21.56 )", "But always prefer the exact form for mathematical rigor."]