\(x^2 + 2x - 168 = 0\).

["# Solve (x^2 + 2x - 168 = 0): Step-by-Step Quadratic Equation Guide", "Solving quadratic equations is a fundamental skill in algebra, useful in various fields from physics to economics. One common equation students and professionals encounter is:", "[\nx^2 + 2x - 168 = 0\n]", "This quadratic equation can be efficiently solved using factoring, the quadratic formula, or completing the square. In this article, we’ll explore all three methods to find the precise solutions, explain key concepts, and highlight practical applications.", "---", "## Why Solve (x^2 + 2x - 168 = 0)?", "Before diving into the solution, understanding why this equation matters adds context. Quadratic equations model real-world phenomena like projectile motion, profit territories, and optimization problems. Solving for (x) helps determine critical points, break-even dates, or measurement data.", "---", "## Method 1: Factoring the Quadratic Equation", "Factoring transforms the equation into a product of binomials, making it easy to identify roots. Let’s rewrite (x^2 + 2x - 168 = 0) in factored form.", "We look for two numbers that:\n- Multiply to (-168) (the constant term),\n- Add to (2) (the coefficient of (x)).", "After testing factor pairs of (-168), we find:\n[\n(x + 14)(x - 12) = 0\n]", "Set each factor to zero:\n[\nx + 14 = 0 \quad \Rightarrow \quad x = -14\n]\n[\nx - 12 = 0 \quad \Rightarrow \quad x = 12\n]", "Solutions:\n[\n\boxed{x = -14} \quad \ ext{and} \quad \boxed{x = 12}\n]", "---", "## Method 2: Apply the Quadratic Formula", "When factoring is difficult, the quadratic formula provides a reliable solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For (ax^2 + bx + c = 0), coefficients are:\n(a = 1), (b = 2), (c = -168).", "### Step 1: Compute the discriminant\n[\n\Delta = b^2 - 4ac = 2^2 - 4(1)(-168) = 4 + 672 = 676\n]", "### Step 2: Take the square root\n[\n\sqrt{676} = 26\n]", "### Step 3: Plug into the formula\n[\nx = \frac{-2 \pm 26}{2(1)} = \frac{-2 \pm 26}{2}\n]", "### Step 4: Solve for both roots\n[\nx = \frac{-2 + 26}{2} = \frac{24}{2} = 12\n]\n[\nx = \frac{-2 - 26}{2} = \frac{-28}{2} = -14\n]", "Solutions (same as factoring):\n[\n\boxed{x = -14}, \quad \boxed{x = 12}\n]", "---", "## Method 3: Completing the Square", "This algebraic technique rewrites the equation into a perfect square trinomial, making extraction of roots straightforward.", "### Step 1: Rewrite the equation\n[\nx^2 + 2x = 168\n]", "### Step 2: Complete the square\nTake half of the coefficient of (x), square it, and add to both sides:\nHalf of 2 is 1; (1^2 = 1).", "[\nx^2 + 2x + 1 = 168 + 1\n]\n[\n(x + 1)^2 = 169\n]", "### Step 3: Solve for (x)\nTake the square root of both sides:\n[\nx + 1 = \pm \sqrt{169} = \pm 13\n]", "[\nx = -1 \pm 13\n]", "### Step 4: Final solutions\n[\nx = -1 + 13 = 12, \quad x = -1 - 13 = -14\n]", "Same results confirmed:\n[\n\boxed{x = -14}, \quad \boxed{x = 12}\n]", "---", "## Verification: Plugging Solutions Back In", "Verify both roots by substituting into the original equation.", "### For (x = 12):\n[\n12^2 + 2(12) - 168 = 144 + 24 - 168 = 0 \quad \checkmark\n]", "### For (x = -14):\n[\n(-14)^2 + 2(-14) - 168 = 196 - 28 - 168 = 0 \quad \checkmark\n]", "Both solutions satisfy the equation.", "---", "## Real-World Applications", "Understanding how to solve (x^2 + 2x - 168 = 0) enables applications such as:", "- Business: Finding break-even points when revenue and cost functions are quadratic.\n- Engineering: Determining maximum stress or deflection in parabolic structures.\n- Physics: Solving for time or position in motion models.", "---", "## Summary", "The equation (x^2 + 2x - 168 = 0) has two real solutions:", "[\n\boxed{x = -14} \quad \ ext{and} \quad \boxed{x = 12}\n]", "Mastering factoring, the quadratic formula, and completing the square not only solves this problem but builds a strong mathematical foundation applicable across sciences and engineering.", "---", "## Key Takeaways", "- Always double-check solutions by substitution.\n- Choose the method that best fits your comfort level—factoring is fastest when possible, the quadratic formula works universally.\n- Quadratic equations model real-life nonlinear relationships.", "Start practicing today—your next problem might just be another quadratic!"]