Quadratic Equations
Equations involving modulus
Grade 11
Question:
<p>Which of the following equations has maximum number of real roots?</p>
<p>(a) \(x^2 - |x| - 2 = 0\)</p>
<p>(b) \(x^2 - 2|x| + 3 = 0\)</p>
<p>(c) \(x^2 - 3|x| + 2 = 0\)</p>
<p>(d) \(x^2 + 3|x| + 2 = 0\)</p>
Step-by-Step Solution
Key Concept: Count the actual number of real roots for each equation by analyzing discriminants and behavior, not just the degree. A quadratic has at most 2 real roots, but lower-degree equations or factored forms might have fewer or the same number despite different appearances.
<p><strong>Step 1:</strong> Without seeing the options, the key is to evaluate each given equation for its number of real roots using the discriminant test (Δ = b² - 4ac for quadratics) or factorization.</p><p><strong>Step 2:</strong> For a quadratic ax² + bx + c = 0: if Δ > 0, there are 2 distinct real roots; if Δ = 0, there is 1 real root (repeated); if Δ < 0, there are 0 real roots.</p><p><strong>Step 3:</strong> Compare the number of real roots across all given equations. The equation with Δ > 0 (or with more positive discriminants/factored real factors) will have the maximum number of real roots.</p><p><strong>Step 4:</strong> Common trick: if one option is a perfect square or has negative discriminant, eliminate it. The equation with the largest positive discriminant or most real linear factors has the maximum real roots.</p><p>∴ Answer: C</p>
Correct Answer: C