\[ n(n + 2) = 210 \] - United Radiology

["# Solve the Equation: ( n(n + 2) = 210 ) – Step-by-Step Guide", "If you’re looking to solve the equation ( n(n + 2) = 210 ), you’re not alone—many students and math enthusiasts tackle this algebraic problem every day. Whether you're preparing for a math test, solving a word problem, or exploring quadratic equations, understanding how to solve ( n(n + 2) = 210 ) is a valuable skill.", "In this SEO-optimized article, we’ll break down step-by-step how to solve the equation ( n(n + 2) = 210 ), explain the underlying math, and explore real-world applications. We’ll also include helpful keywords for search engines such as solve quadratic equations, equation with multiplication, algebraic word problems, and more.", "---", "## What Is the Equation ( n(n + 2) = 210 )?", "The expression ( n(n + 2) = 210 ) represents a quadratic equation in disguise. Expanding it gives:", "[
\nn^2 + 2n = 210
\n]", "Subtracting 210 from both sides leads to:", "[
\nn^2 + 2n - 210 = 0
\n]", "This is now in the standard quadratic form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = 2 ), and ( c = -210 ).", "---", "## Step-by-Step Solution: How to Solve ( n(n + 2) = 210 )", "### Step 1: Expand and Rearrange the Equation", "Start by expanding the left-hand side:", "[
\nn^2 + 2n = 210
\n]", "Subtract 210 to bring the equation to standard quadratic form:", "[
\nn^2 + 2n - 210 = 0
\n]", "### Step 2: Factor the Quadratic (If Possible)", "We look for two numbers that multiply to (-210) and add to (2). These numbers are (15) and (-14), since:", "[
\n15 \ imes (-14) = -210 \quad \ ext{and} \quad 15 + (-14) = 1
\n]", "Oops! That adds to 1, not 2. Let’s double-check factors of 210 that differ by 2.", "Actually, try (15) and (-14) rechecked: They do add to 1, so not correct.", "Try another pair: (14) and (-15) → sum is (-1)", "Wait — actually, correct pairs multiplying to (-210) and adding to (2):", "We try 15 and −14 → sum = 1 → no", "14 × (−15) = −210, sum = −1
\n16 × (−13.125) — not integer", "Better approach: Use the quadratic formula since factoring is tricky.", "---", "### Step 3: Apply the Quadratic Formula", "For ( n^2 + 2n - 210 = 0 ), the quadratic formula is:", "[
\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plug in ( a = 1 ), ( b = 2 ), ( c = -210 ):", "[
\nn = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-210)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm \sqrt{844}}{2}
\n]", "Now simplify ( \sqrt{844} ):", "Factor 844:
\n( 844 = 4 \ imes 211 ) → so
\n[
\n\sqrt{844} = \sqrt{4 \ imes 211} = 2\sqrt{211}
\n]", "Thus,", "[
\nn = \frac{-2 \pm 2\sqrt{211}}{2} = -1 \pm \sqrt{211}
\n]", "Since ( \sqrt{211} \approx 14.525 ), we get two solutions:", "[
\nn \approx -1 + 14.525 = 13.525 \quad \ ext{and} \quad n \approx -1 - 14.525 = -15.525
\n]", "---", "## Are There Real Integer Solutions?", "Let’s test nearby integers to see if ( n(n + 2) = 210 ) has clean answers.", "Try ( n = 13 ):", "[
\n13 \ imes 15 = 195
\n]", "Try ( n = 14 ):
\n[
\n14 \ imes 16 = 224 \quad (\ ext{too big})
\n]", "Try ( n = 15 ):
\n[
\n15 \ imes 17 = 255 \quad (\ ext{too big})
\n]", "No integer solution satisfies the equation exactly — the roots are irrational. So the exact solutions are:", "[
\nn = -1 + \sqrt{211} \quad \ ext{and} \quad n = -1 - \sqrt{211}
\n]", "---", "## Practical Tips: Solving Quadratic Equations Like This", "- Double-check expansion: Always expand expressions before rearranging.
\n- Use the quadratic formula when factoring isn’t obvious.
\n- Estimate roots using square roots of nearby perfect squares.
\n- Check solutions by plugging back into the original equation.
\n- Applications: Such equations appear in projectile motion, area problems, and optimization.", "---", "## Real-World Context: Why This Equation Matters", "Solving equations like ( n(n + 2) = 210 ) helps model situations where the product of two consecutive quantities equals a known total. For example:", "- Land division: If a plot has length ( n ) and width ( n+2 ), and area = 210 m², find ( n ).
\n- Age problems: Related to time differences or consecutive measurements.
\n- Financial math: Modeling revenue arrays where dimensions increase.", "---", "## Frequently Asked Questions (FAQ)", "### Q: Can ( n(n + 2) = 210 ) have integer solutions?
\nA: This specific equation has no integer solutions. The closest products are 195 (13×15) and 224 (14×16).", "### Q: What’s the best way to solve quadratic equations with no neat factors?
\nA: Use the quadratic formula for accuracy.", "### Q: How do I verify my solution?
\nA: Substitute ( n ) back into ( n(n + 2) ); it should equal 210.", "---", "## Conclusion", "The equation ( n(n + 2) = 210 ) leads to a quadratic that doesn’t factor easily, but solving it exactly uses the quadratic formula:", "[
\nn = -1 \pm \sqrt{211}
\n]", "While no integer solution exists, understanding how to solve such equations strengthens algebraic fluency and problem-solving for real-life scenarios.", "---", "### Key SEO Keywords:
\n- Solve quadratic equations
\n- Quadratic formula应用实例
\n- Solve ( n(n + 2) = 210 )
\n- Equation with multiplication
\n- Algebra word problems
\n- Quadratic factoring
\n- Real-world algebra applications", "---", "### Final Thought", "Mastering equations like ( n(n + 2) = 210 ) is more than academic — it’s about building analytical thinking and problem-solving skills essential in STEM fields, finance, engineering, and everyday decision-making. Keep practicing, and you’ll master quadratics in no time!", "---", "Meta Title: Solve ( n(n + 2) = 210 ) — Step-by-Step Guide with Solutions
\nMeta Description: Learn how to solve ( n(n + 2) = 210 ) using algebra, factoring, and the quadratic formula. Includes solutions and real-world applications.
\nTags: #SolveQuadratics #Algebra #QuadraticEquations #MathTutorial #EMSolutions #EquationExamples"]