Quadratic Equations
Common roots
Grade 11
Question:
<p>If <em>α</em> and <em>β</em>, <em>α</em> and <em>γ</em>, <em>α</em> and <em>δ</em> are the roots of the equations \(ax^2 + 2bx + c = 0\), \(2bx^2 + cx + a = 0\) and \(cx^2 + ax + 2b = 0\), respectively, where <em>a</em>, <em>b</em> and <em>c</em> are positive real numbers, then \(\alpha + \alpha^2 =\)</p>
<p>\(abc\)</p>
<p>\(a + 2b + c\)</p>
<p>\(-1\)</p>
<p>\(0\)</p>
Step-by-Step Solution
Key Concept: Since α appears as a common root in all three equations, it must simultaneously satisfy all three equations. Subtract pairs of equations to eliminate α and create a relationship that reveals α's value independent of the coefficients.
<p><strong>Step 1:</strong> Since α is a root of all three equations:</p><p>aα² + 2bα + c = 0 ... (1)</p><p>2bα² + cα + a = 0 ... (2)</p><p>cα² + aα + 2b = 0 ... (3)</p><p><strong>Step 2:</strong> Add all three equations:</p><p>(a + 2b + c)α² + (2b + c + a)α + (c + a + 2b) = 0</p><p>(a + 2b + c)(α² + α + 1) = 0</p><p><strong>Step 3:</strong> Since a, b, c > 0, we have a + 2b + c > 0</p><p>Therefore: α² + α + 1 = 0</p><p><strong>Step 4:</strong> This gives us: α + α² = -1</p><p>∴ Answer: C (which equals -1)</p>
Correct Answer: C