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

["# 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."]