x = \frac{-35 \pm \sqrt{35^2 - 4 \times 2 \times (-175)}}{4}

["Solving the Quadratic Equation: A Step-by-Step Guide to ( x = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes (-175)}}{4} )", "Quadratic equations form the foundation of algebra and are essential in many areas of mathematics, physics, and engineering. One such quadratic equation presented here is:", "$$\nx = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes (-175)}}{4}\n$$", "This expression provides the exact solutions (roots) of the quadratic equation ( 2x^2 + 35x + 175 = 0 ) using the quadratic formula. In this article, we’ll walk through how to simplify and evaluate this equation, explain each component, and explore how to interpret these roots.", "---", "### Understanding the Quadratic Formula", "The general form of a quadratic equation is:", "$$\nax^2 + bx + c = 0\n$$", "The solutions are given by:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "In our case:\n- ( a = 2 )\n- ( b = 35 )\n- ( c = 175 )", "Plugging these values into the formula, we get:", "$$\nx = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes 175}}{2 \ imes 2}\n$$", "Simplifying the denominator:\n( 2 \ imes 2 = 4 )", "So the equation becomes:", "$$\nx = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes 175}}{4}\n$$", "This matches the given expression.", "---", "### Step 1: Compute the Discriminant", "The part under the square root is called the discriminant, denoted as ( D = b^2 - 4ac ). It determines the nature of the roots:", "$$\nD = 35^2 - 4 \ imes 2 \ imes (-175)\n$$", "Calculate each term:", "- ( 35^2 = 1225 )\n- ( 4 \ imes 2 \ imes (-175) = -1400 ), but since it’s subtracted in discriminant, it becomes ( -(-1400) = +1400 )", "Thus:", "$$\nD = 1225 + 1400 = 2625\n$$", "---", "### Step 2: Simplify the Square Root", "We now have:", "$$\n\sqrt{2625}\n$$", "Let’s factor 2625 to simplify:", "- ( 2625 \div 25 = 105 )\n- ( 105 = 3 \ imes 5 \ imes 7 )", "So:", "$$\n2625 = 25 \ imes 105 = 5^2 \ imes 3 \ imes 5 \ imes 7 = 3 \ imes 5^2 \ imes 7\n$$", "Thus:", "$$\n\sqrt{2625} = \sqrt{25 \ imes 105} = 5\sqrt{105}\n$$", "So the discriminant simplifies to:", "$$\n\sqrt{2625} = 5\sqrt{105}\n$$", "---", "### Step 3: Writing the Final Roots", "Now substitute back into the quadratic formula:", "$$\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n$$", "We can simplify by factoring numerator:", "$$\nx = \frac{-35}{4} \pm \frac{5\sqrt{105}}{4}\n$$", "Thus, the two solutions are:", "$$\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n$$", "---", "### Step 4: Interpreting the Roots", "The solutions represent two real and distinct roots, since ( D = 2625 > 0 ). These irrational numbers (due to ( \sqrt{105} )) cannot be simplified further as exact decimals, but they are precise solutions.", "In real-world applications, such roots might model projectile paths, profit maximization, or equilibrium points in physics and economics.", "---", "### Conclusion: Mastery of Quadratic Equations", "The expression\n[\nx = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes (-175)}}{4}\n]\nis a perfect example of applying the quadratic formula with careful discriminant evaluation. Simplifying the square root via factoring enhances clarity and approach usage in both academic and applied problem-solving contexts.", "For students and practitioners, mastering this process enables confident handling of quadratic models across science, engineering, and finance.", "---", "### SEO Elements Included:", "- Keyword optimization: Primary keywords like “quadratic equation,” “solve ( x = \frac{-35 \pm \sqrt{\cdots}}{4} )”, “quadratic formula,” and “discriminant” are naturally integrated.\n- Structured content: Clear section headers guide readers and help with on-page SEO.\n- Technical explanation: Step-by-step derivation improves readability and content depth for SEO relevance.\n- Practical insight: Concluding interpretation links theory to real-world application.", "---", "Keywords:\nquadratic equation ( x = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes (-175)}}{4} ), solving quadratic with discriminant, quadratic formula steps, simplifying square roots, exact quadratic solutions, real and complex roots.", "Meta Description:\nLearn how to solve the quadratic equation ( x = \frac{-35 \pm \sqrt{35^2 - 4 \ imes 2 \ imes (-175)}}{4} ) step-by-step, including discriminant analysis, simplification via factoring, and real-world interpretation.", "---", "Keep mastering algebra — the foundations of math everywhere begin with quadratics!"]