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Question
If roots α, β of the equation \[x^2 - px + 16 = 0\] satisfy the relation α2 + β2 = 9, then write the value P.
Solution
Given equation: \[x^2 - px + 16 = 0\]
Also,
\[\alpha\] and \[\beta\] are the roots of the equation satisfying \[\alpha^2 + \beta^2 = 9 .\]
From the equation, we have:
Sum of the roots = \[\alpha + \beta =\]\[- \left( \frac{- p}{1} \right) = p\]
Product of the roots = \[\alpha\beta\] = \[\frac{16}{1} = 16\]
\[\text { Now }, \left( \alpha + \beta \right)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\]
\[ \Rightarrow p^2 = 9 + 32\]
\[ \Rightarrow p^2 = 41\]
\[ \Rightarrow p = \sqrt{41}\]
Hence, the value of \[p \text { is } \sqrt{41} .\]
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