TS EAMCET · Maths · Quadratic Equation
If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^4-4 x^3+3 x^2+2 x-2=0\) such that \(\alpha\) and \(\beta\) are integers and \(\gamma, \delta\) are irrational numbers, then \(\alpha+2 \beta+\gamma^2+\delta^2=\)
- A \(5\)
- B \(7\)
- C \(11\)
- D \(13\)
Answer & Solution
Correct Answer
(C) \(11\)
Step-by-step Solution
Detailed explanation
\(P(x) = x^4-4 x^3+3 x^2+2 x-2\) \(P(1) = 1-4+3+2-2 = 0\) \(P'(x) = 4x^3-12x^2+6x+2\) \(P'(1) = 4-12+6+2 = 0\) Since \(P(1)=0\) and \(P'(1)=0\), \(x=1\) is a root of multiplicity at least 2. So, \(\alpha=1, \beta=1\). Factoring \(P(x)\) by \((x-1)^2 = x^2-2x+1\):…
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