Quadratic Equations
Equations involving absolute values
Grade 11
Question:
<p>When \(x > \frac{3}{2}\), the equation \(x^2 + 2x - 7 = 0\) gives roots \(x = -1 \pm 2\sqrt{2}\). For \(x < \frac{3}{2}\), the equation \(x^2 - 2x - 1 = 0\) gives roots \(x = 1 \pm \sqrt{2}\). The sum of valid roots of \(|x^2 + 2x - 7| = x^2 - 2x + 1\) is:</p>
<p>\(\sqrt{2}\)</p>
<p>\(2\sqrt{2}\)</p>
<p>\(0\)</p>
<p>\(1\)</p>
Step-by-Step Solution
Key Concept: Recognize that a quadratic equation has two distinct roots, and the condition x > 3/2 selects only one specific root. For the complementary domain x < 3/2, you must identify which root satisfies this inequality and express it correctly.
<p><strong>Step 1:</strong> The quadratic x² + 2x - 7 = 0 has two roots: x = -1 + 2√2 and x = -1 - 2√2.</p><p><strong>Step 2:</strong> Evaluate which root satisfies x > 3/2. Since -1 + 2√2 ≈ 1.83 > 1.5, this root satisfies the first condition.</p><p><strong>Step 3:</strong> For the complementary domain x < 3/2, the other root must apply. Check: -1 - 2√2 ≈ -3.83 < 1.5 ✓</p><p><strong>Step 4:</strong> Therefore, when x < 3/2, the solution to x² + 2x - 7 = 0 is x = -1 - 2√2.</p><p>∴ Answer: A</p>
Correct Answer: A