Question

let f(x) be an identity fuction and alpha,beta be the roots of the equation x^2-5x+9=0 then the value of f(alpha)+f(beta) is equal to

Answer 1

$\textbf{Given:}$

$\text{ \alpha and \beta are roots of x^2-5x+9=0 }$

$\text{ f(x) is an identity function}$

$\textbf{To find:}$

$f(\alpha)+f(\beta)$

$\textbf{Solution:}$

$\text{Consider,}\,x^2-5x+9=0$

$x^2-5x=-9$

$x^2-5x+\frac{5}{4}=-9+\frac{5}{4}$

$(x-\frac{5}{2})^2=\frac{-31}{4}$

$x-\frac{5}{2}=\pm\,i\frac{\sqrt{31}}{2}$

$\implies\,x=\frac{5}{2}\pm\,i\frac{\sqrt{31}}{2}$

$\implies\,\alpha=\frac{5}{2}+\,i\frac{\sqrt{31}}{2}\;\text{and}\;\beta=\frac{5}{2}-\,i\frac{\sqrt{31}}{2}$

$\text{Since f(x) is an identity function, we have}$

$f(\alpha)+f(\beta)$

$=\alpha+\beta$

$=\frac{5}{2}+i\frac{\sqrt{31}}{2}+\frac{5}{2}-i\frac{\sqrt{31}}{2}$

$=\frac{5}{2}+\frac{5}{2}$

$=5$

$\therefore\textbf{The value of \bf\,f(\alpha)+f(\beta) is 5}$

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