AP EAMCET · Maths · Quadratic Equation
The difference of the irrational roots of the equation \(x^5-5 x^4+9 x^3-9 x^2+5 x-1=0\) is
- A \(\sqrt{3}\)
- B \(2 \sqrt{5}\)
- C \(3\)
- D \(\sqrt{5}\)
Answer & Solution
Correct Answer
(D) \(\sqrt{5}\)
Step-by-step Solution
Detailed explanation
\(P(x) = x^5-5 x^4+9 x^3-9 x^2+5 x-1\) \(P(1) = 1-5+9-9+5-1 = 0\). Thus, \(x=1\) is a root. \((x^5-5 x^4+9 x^3-9 x^2+5 x-1) \div (x-1) = x^4-4x^3+5x^2-4x+1 = 0\) Divide by \(x^2\): \(x^2-4x+5-4/x+1/x^2 = 0\) \((x^2+1/x^2) - 4(x+1/x) + 5 = 0\) Let \(y=x+1/x\), then…
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