\[ x^2 + 2x - 135 = 0 \]

["# Solving the Quadratic Equation ( x^2 + 2x - 135 = 0 ): A Step-by-Step Guide", "Quadratic equations form a foundational concept in algebra, offering essential tools for solving a wide range of real-world problems—from physics to engineering. One commonly encountered equation is:", "[\nx^2 + 2x - 135 = 0\n]", "Understanding how to solve this equation not only helps in mastering algebra but also builds confidence in tackling more complex quadratic expressions. In this SEO-optimized article, we’ll explore how to solve ( x^2 + 2x - 135 = 0 ) using multiple methods, including factoring, completing the square, and the quadratic formula. We’ll also include practical tips for better comprehension and easier learning.", "---", "## Why Are Quadratic Equations Important?", "Quadratic equations are essential because they model parabolic relationships, useful in projectile motion, profit maximization, geometry, and optimization problems. Knowing how to solve ( x^2 + 2x - 135 = 0 ) exemplifies key algebraic techniques applicable across disciplines.", "---", "## Understanding the Equation", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "Compare with your equation:\n( a = 1 ), ( b = 2 ), ( c = -135 )", "---", "## Method 1: Factoring the Quadratic Expression", "Factoring is often the quickest method when the trinomial is easily factorable.", "### Step 1: Identify coefficients\nWe want two numbers that multiply to ( a \cdot c = 1 \cdot (-135) = -135 ) and add to ( b = 2 ).", "### Step 2: Find the factors\nWhich two integers multiply to -135 and add to 2?\nTry ( 13 ) and ( -11 ):\n( 13 \ imes (-11) = -135 ), ( 13 + (-11) = 2 ) — perfect!", "### Step 3: Write the factored form\n[\nx^2 + 2x - 135 = (x + 13)(x - 11) = 0\n]", "### Step 4: Solve using the zero-product property\nSet each factor equal to zero:\n[\nx + 13 = 0 \Rightarrow x = -13\n]\n[\nx - 11 = 0 \Rightarrow x = 11\n]", "---", "## Method 2: Completing the Square", "Completing the square is a powerful technique useful in graphing and deeper understanding.", "### Step 1: Move constant term\n[\nx^2 + 2x = 135\n]", "### Step 2: Complete the square\nTake half of coefficient of ( x ), which is ( 2/2 = 1 ), then square it: ( 1^2 = 1 ).\nAdd 1 to both sides:\n[\nx^2 + 2x + 1 = 135 + 1\n\Rightarrow (x + 1)^2 = 136\n]", "### Step 3: Take square roots\n[\nx + 1 = \pm \sqrt{136} = \pm 2\sqrt{34}\n]", "### Step 4: Solve for ( x )\n[\nx = -1 \pm 2\sqrt{34}\n]", "---", "## Method 3: Using the Quadratic Formula", "The quadratic formula provides a universal solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 1: Plug in values\n( a = 1 ), ( b = 2 ), ( c = -135 )", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-135)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 540}}{2} = \frac{-2 \pm \sqrt{544}}{2}\n]", "### Step 2: Simplify the square root\n( \sqrt{544} = \sqrt{16 \cdot 34} = 4\sqrt{34} )", "[\nx = \frac{-2 \pm 4\sqrt{34}}{2} = -1 \pm 2\sqrt{34}\n]", "---", "## Final Solutions", "Whether by factoring, completing the square, or the quadratic formula, the solutions are the same:", "[\nx = -13 \quad \ ext{and} \quad x = 11\n]", "These are the roots of the equation ( x^2 + 2x - 135 = 0 ), representing points where the parabola intersects the x-axis.", "---", "## Practical Tips for Mastery", "- Pre-factor constant term for speed: When factoring, look for pairs of numbers quickly—practice helps!\n- Verify solutions: Plug ( x = -13 ) and ( x = 11 ) back into the original equation to confirm.\n- Visualize the graph: The solutions are the x-intercepts; use graphing tools to reinforce understanding.\n- Apply context: Imagine solving for when a ball hits the ground (time = ( x )) using physics models.", "---", "## Conclusion", "Solving ( x^2 + 2x - 135 = 0 ) demonstrates key algebraic skills—factoring, completing the square, and applying the quadratic formula—each offering a unique perspective. Mastering these methods strengthens problem-solving abilities and prepares learners for advanced math and real-world applications.", "Keywords: quadratic equation solution, solve (x^2 + 2x - 135 = 0), factoring quadratic, completing the square, quadratic formula, algebra learning, quadratic roots, solving equations.", "---", "Want to go deeper? Explore our guides on quadratic inequalities, discriminant analysis, and applications of parabolas in science and finance."]