<p><strong>708.</strong> Let \( f(x) = x^2 + \alpha x + \beta \) where \( \alpha, \beta \in R \) and \( f(f(x)) = 0 \) has 2 roots 1 and 2. Find the value of \( 2|f(0)| \).</p>
Step-by-Step Solution
Key Concept: If f(f(x)) = 0 has roots 1 and 2, then f(1) and f(2) must be roots of f(t) = 0. Use this relationship along with Vieta's formulas to find α and β.
<p><strong>Step 1:</strong> Let the roots of f(t) = 0 be r and s. Then f(t) = (t - r)(t - s).</p><p><strong>Step 2:</strong> Since f(f(x)) = 0 has roots x = 1 and x = 2, we have f(1) and f(2) ∈ {r, s}.</p><p><strong>Step 3:</strong> Calculate f(1) = 1 + α + β and f(2) = 4 + 2α + β.</p><p><strong>Step 4:</strong> By Vieta's formulas for f(t) = 0: r + s = -α and rs = β.</p><p><strong>Step 5:</strong> By Vieta's formulas for f(f(x)) = 0: f(1) + f(2) = -(sum of roots of f) = α. This gives: (1 + α + β) + (4 + 2α + β) = α, so 5 + 3α + 2β = α, thus 2α + 2β = -5.</p><p><strong>Step 6:</strong> Product: f(1)·f(2) = (1 + α + β)(4 + 2α + β) = β. From sum relation: 2α + 2β = -5, so α + β = -5/2.</p><p><strong>Step 7:</strong> Substituting β = -5/2 - α into (1 + α + β)(4 + 2α + β) = β: (1 - 5/2)(4 + 2α - 5/2 - α) = -5/2 - α, which gives (-3/2)(3/2 + α) = -5/2 - α.</p><p><strong>Step 8:</strong> Solving: -9/4 - 3α/2 = -5/2 - α, so -α/2 = 1/4, thus α = -1/2.</p><p><strong>Step 9:</strong> Then β = -5/2 - (-1/2) = -2.</p><p><strong>Step 10:</strong> Therefore f(0) = β = -2, and |f(0)| = 2.</p><p>∴ Answer: 2|f(0)| = <strong>4</strong></p>
Correct Answer: 2