["# Solve ( x^2 + (x + 2)^2 = 340 ): Step-by-Step Guide to Find All Real Solutions", "If you're looking to solve the quadratic equation ( x^2 + (x + 2)^2 = 340 ), you're not alone. This type of equation appears frequently in algebra and competitive math problems. In this comprehensive article, we’ll walk you through the step-by-step process of solving ( x^2 + (x + 2)^2 = 340 ), explain the underlying math, and help you understand how to apply analogous methods to similar problems.", "---", "## What Is the Equation?", "Start with the given equation:", "[
\nx^2 + (x + 2)^2 = 340
\n]", "This equation involves the sum of a perfect square trinomial and a quadratic term. Expanding and simplifying leads us to a standard quadratic form.", "---", "## Step 1: Expand the Equation", "Begin by expanding ( (x + 2)^2 ):", "[
\n(x + 2)^2 = x^2 + 4x + 4
\n]", "Now replace into the original equation:", "[
\nx^2 + (x^2 + 4x + 4) = 340
\n]", "Combine like terms:", "[
\n2x^2 + 4x + 4 = 340
\n]", "---", "## Step 2: Simplify to Standard Quadratic Form", "Subtract 340 from both sides:", "[
\n2x^2 + 4x + 4 - 340 = 0
\n]", "[
\n2x^2 + 4x - 336 = 0
\n]", "To make solving easier, divide the entire equation by 2:", "[
\nx^2 + 2x - 168 = 0
\n]", "---", "## Step 3: Solve the Quadratic Equation", "Use the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "For ( x^2 + 2x - 168 = 0 ), coefficients are:", "- ( a = 1 )
\n- ( b = 2 )
\n- ( c = -168 )", "Plug in values:", "[
\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-168)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 672}}{2} = \frac{-2 \pm \sqrt{676}}{2}
\n]", "Since ( \sqrt{676} = 26 ), we get:", "[
\nx = \frac{-2 \pm 26}{2}
\n]", "---", "## Step 4: Compute the Two Solutions", "[
\nx = \frac{-2 + 26}{2} = \frac{24}{2} = 12
\n]", "[
\nx = \frac{-2 - 26}{2} = \frac{-28}{2} = -14
\n]", "---", "## Step 5: Verify the Solutions", "Check ( x = 12 ):", "[
\n12^2 + (12 + 2)^2 = 144 + 196 = 340 \quad \ ext{✓}
\n]", "Check ( x = -14 ):", "[
\n(-14)^2 + (-14 + 2)^2 = 196 + (-12)^2 = 196 + 144 = 340 \quad \ ext{✓}
\n]", "Both solutions satisfy the original equation.", "---", "## Additional Insights", "### Why Expanding Works", "Expanding ( (x + 2)^2 ) converts a sum of squares into a simple quadratic. This standard form allows direct application of the quadratic formula or factorization.", "### Alternative Approach: Completing the Square", "You could also solve without full expansion by expanding first, then completing the square on ( 2x^2 + 4x ), factoring out 2, and completing the square. This reinforces different problem-solving strategies.", "---", "## Common Pitfalls to Avoid", "- Forgetting to expand fully before combining terms
\n- Errors in sign handling when substituting and simplifying
\n- Misapplying the quadratic formula due to missed coefficients
\n- Skipping verification, leading to undetected mistakes", "---", "## Real-World Applications", "Equations like ( x^2 + (x + 2)^2 = 340 ) often appear in physics (e.g., projectile motion or energy calculations), engineering problems involving geometric shapes, or optimization tasks where symmetric components contribute to a total.", "---", "## Summary of Key Solutions", "The real solutions to ( x^2 + (x + 2)^2 = 340 ) are:", "- ( x = 12 )
\n- ( x = -14 )", "These values arise from solving a symmetric quadratic equation derived from symmetric expressions.", "---", "## Further Learning", "To master equations like this:", "- Practice expanding binomials and simplifying expressions
\n- Learn to manipulate and solve quadratic equations by multiple methods
\n- Explore graphing quadratic functions to visualize symmetry and solutions", "Understanding this problem enhances your algebraic toolkit and prepares you for more complex equations in advanced math.", "---", "## Keyword Summary (SEO Focus)", "- Solve ( x^2 + (x + 2)^2 = 340 )
\n- Quadratic equation solution
\n- Step-by-step algebra
\n- Expand and solve
\n- Real solutions to symmetric quadratics
\n- Algebra problem tutorial
\n- Math practice equation", "---", "Start mastering quadratic equations today — understanding equations like ( x^2 + (x + 2)^2 = 340 ) opens doors to deeper mathematical insight!"]
