Solve the quadratic equation \( n^2 + n - 10100 = 0 \) using the quadratic formula \( n = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
["# Solving the Quadratic Equation ( n^2 + n - 10100 = 0 ) Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, widely used in science, engineering, and real-world applications. In this article, we’ll solve the equation ( n^2 + n - 10100 = 0 ) step by step using the quadratic formula. Whether you're a student, teacher, or self-learner, understanding how to solve such equations will strengthen your problem-solving toolkit.", "## What is a Quadratic Equation?", "A quadratic equation has the standard form:", "[\nan^2 + bn + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). The solutions for ( n ) are given by the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula provides exact values for the roots, whether real or complex.", "## Step-by-Step Solution of ( n^2 + n - 10100 = 0 )", "Our given equation is:", "[\nn^2 + n - 10100 = 0\n]", "Here, the coefficients are:\n- ( a = 1 )\n- ( b = 1 )\n- ( c = -10100 )", "### Step 1: Plug values into the quadratic formula", "Substituting ( a ), ( b ), and ( c ) into the formula:", "[\nn = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-10100)}}{2(1)}\n]", "### Step 2: Simplify inside the square root", "[\nn = \frac{-1 \pm \sqrt{1 + 40400}}{2}\n]", "[\nn = \frac{-1 \pm \sqrt{40401}}{2}\n]", "### Step 3: Compute the square root", "We need to find ( \sqrt{40401} ). Recall that:", "[\n201^2 = 201 \ imes 201 = 40401\n]", "Thus,", "[\n\sqrt{40401} = 201\n]", "### Step 4: Substitute back", "[\nn = \frac{-1 \pm 201}{2}\n]", "This gives two possible solutions:", "[\nn = \frac{-1 + 201}{2} = \frac{200}{2} = 100\n]", "[\nn = \frac{-1 - 201}{2} = \frac{-202}{2} = -101\n]", "## Final Solutions", "The two real solutions to the equation ( n^2 + n - 10100 = 0 ) are:", "[\nn = 100 \quad \ ext{and} \quad n = -101\n]", "## Applications of the Equation", "This particular equation models scenarios involving area or motion where a quadratic relationship arises. For example, it could represent:", "- The width and length of a rectangular field with area and side constraints,\n- The timing of motion with a quadratic displacement function.", "Understanding how to solve such equations enables deeper insight and practical problem-solving in physics, economics, and computer science.", "## Conclusion", "Using the quadratic formula provides a reliable way to find exact solutions to quadratic equations. For ( n^2 + n - 10100 = 0 ), applying the formula confirms the solutions ( n = 100 ) and ( n = -101 ). Mastering this method enhances your algebraic proficiency and prepares you for advanced mathematical challenges.", "---", "Keywords: quadratic equation solution, quadratic formula, solve ( n^2 + n - 10100 = 0 ), step-by-step algebra, real roots of quadratic, analyze parabola, algebra tutorial, mathematical problem solving", "Meta Description: Solve the quadratic equation ( n^2 + n - 10100 = 0 ) using the quadratic formula with clear steps and real examples. Learn how to apply ( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) for fast, accurate results."]
