Question

If roots α, β of the equation \[x^2 - px + 16 = 0\] satisfy the relation α² + β² = 9, then write the value P.

Answer 1

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 \[p \text { is } \sqrt{41} .\]