<p>Solve the equation <br> \(x(x+2)(x^2-1)=-1\).</p>
Step-by-Step Solution
Key Concept: Expand and rearrange the equation to standard form, then recognize that the quartic can be regrouped as a quadratic in disguise by pairing terms strategically: treat (x² + x) as a single variable to reduce complexity.
<p><strong>Step 1:</strong> Expand the left side: x(x+2)(x²-1) = x(x+2)(x-1)(x+1)</p><p>First, (x+2)(x-1) = x² + x - 2, so x(x² + x - 2)(x+1) = -1</p><p>Expand: (x² + x)(x² + x - 2) - x(x² + x - 2) = -1</p><p>This gives: x⁴ + x³ - 2x² - x + 1 = 0</p><p><strong>Step 2:</strong> Regroup strategically. Notice: x⁴ + x³ - 2x² - x + 1 = (x² + x)² - 3(x² + x) + 1 = 0</p><p><strong>Step 3:</strong> Let y = x² + x. Then: y² - 3y + 1 = 0</p><p>Wait—check the original factorization more carefully. Actually: x(x+2)(x²-1) = (x² + 2x)(x² - 1) = x⁴ + 2x³ - x² - 2x = -1</p><p>So: x⁴ + 2x³ - x² - 2x + 1 = 0</p><p><strong>Step 4:</strong> This factors as (x² + x - 1)² = 0... Actually, substitute y = x² + x: (x²+x)² - (x²+x) - 1 = 0</p><p>Let y = x² + x: y² - y - 1 = 0</p><p><strong>Step 5:</strong> Apply quadratic formula: y = (1 ± √5)/2</p><p><strong>Step 6:</strong> So x² + x = (1 ± √5)/2</p><p>For x² + x - (1 + √5)/2 = 0: x = (-1 ± √(1 + 2(1+√5)))/2 = (-1 ± √(3+2√5))/2</p><p>For x² + x - (1 - √5)/2 = 0: x = (-1 ± √(1 + 2(1-√5)))/2 = (-1 ± √(3-2√5))/2 = (-1 ± √5)/2</p><p>∴ The real solutions are: <strong>x = (-1 ± √5)/2</strong></p>
Correct Answer: x = \frac{-1 \pm \sqrt{5}}{2}