Question

Find the quadratic polynomial whose the sum and product of the zeros are one upon root 2 + 1 and root 2 + 1

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\:\:$

$\green{\underline \bold{Given :}}$

$\:\:$

• Sum of zeros is $\rm \dfrac { 1 } { \sqrt 2 + 1 }$

$\:\:$

• Product of zeros is $\rm \sqrt 2 + 1$

$\:\:$

$\red{\underline \bold{To \: Find:}}$

$\:\:$

• The required quadratic polynomial.

$\:\:$

$\large{\orange{\underline{\tt{Solution :-}}}}$

$\:\:$

$\sf Let \: \alpha \: \& \: \beta \: be \: the \: zeroes \: of \: quadratic \: polynomial.$

$\:\:$

$\underline{\bold{\texttt{Quadratic polynomial will be:}}}$

$\:\:$

$\sf \implies x^2 - (\alpha + \beta)x + \alpha\beta$ -------(1)

$\:\:$

$\underline{\bold{\texttt{We are given :}}}$

$\:\:$

$\purple\longrightarrow$ $\rm \alpha + \beta = \dfrac { 1 } { \sqrt 2 + 1 }$

$\:\:$

$\purple\longrightarrow$ $\rm \alpha\beta = \sqrt 2 + 1$

$\:\:$

$\underline{\bold{\texttt{Rationalizing}}}$ $\sf \dfrac { 1 } { \sqrt 2 + 1 }$

$\:\:$

Multiplying numerator amd denominator by $\rm \sqrt 2 - 1$

$\:\:$

$\sf \longmapsto \dfrac { 1 } { \sqrt 2 + 1 } \times \dfrac { \sqrt 2 - 1 } { \sqrt 2 - 1 }$

$\:\:$

$\sf \longmapsto \dfrac { \sqrt 2 - 1 } { 2 - 1 }$

$\:\:$

$\bf \dashrightarrow \sqrt 2 - 1$

$\:\:$

$\underline{\bold{\texttt{We got:}}}$

$\:\:$

$\rm \alpha + \beta = \sqrt 2 - 1$

$\:\:$

$\underline{\bold{\texttt{Putting the values in (1)}}}$

$\:\:$

$\sf \longmapsto x^2 - (\sqrt 2 - 1)x + \sqrt 2 + 1$

$\:\:$

$\underline{\bold{\texttt{Required quadratic polynomial is:}}}$

$\:\:$

$\bf x^2 - \sqrt 2x + 1x + \sqrt 2 + 1$

$\rule{200}5$