Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
If $α$ and $β$ are the zeros of the quadratic polynomial $f(x)\ =\ x^2\ -\ 3x\ -\ 2$, find a quadratic polynomial whose zeros are $\frac{1}{2α\ +\ β}$ and $\frac{1}{2β\ +\ α}$.
Given:
$α$ and $β$ are the zeros of the quadratic polynomial $f(x)\ =\ x^2\ -\ 3x\ -\ 2$.
To do:
We have to find the quadratic polynomial having $\frac{1}{2α+β}$ and $\frac{1}{2β+α}$ as its zeros.
Solution:
We know that,
The standard form of a quadratic equation is $ax^2+bx+c=0$, where a, b and c are
constants and $a≠0$
Comparing the given equation with the standard form of a quadratic equation,
$a=1$, $b=-3$ and $c=-2$
Sum of the roots $= α+β = \frac{-b}{a} = \frac{-(-3)}{1}=3$.
Product of the roots $= αβ = \frac{c}{a} = \frac{-2}{1}=-2$.
Let the sum and product of the zeros of the given quadratic equation be $S$ and $P$.
Therefore,
$S=\frac{1}{2\alpha +\beta } +\frac{1}{2\beta +\alpha }$
$=\frac{2\beta +\alpha +2\alpha +\beta }{( 2\alpha +\beta )( 2\beta +\alpha )}$
$=\frac{3\alpha +3\beta }{4\alpha \beta +2\alpha ^{2} +2\beta ^{2} +\alpha \beta }$
$=\frac{3( \alpha +\beta )}{5\alpha \beta +2\left( \alpha ^{2} +\beta ^{2}\right)}$
$=\frac{3( \alpha +\beta )}{5\alpha \beta +2\left(( \alpha +\beta )^{2} -2\alpha \beta \right)}$
$=\frac{3( 3)}{5( -2) +2\left(( 3)^{2} -2( -2)\right)}$
$=\frac{9}{-10+2( 9+4)}$
$=\frac{9}{-10+26}$
$=\frac{9}{16}$
$P=\frac{1}{2\alpha +\beta } \times \frac{1}{2\beta +\alpha }$
$=\frac{1}{( 2\alpha +\beta )( 2\beta +\alpha )}$
$=\frac{1}{4\alpha \beta +2\alpha ^{2} +2\beta ^{2} +\alpha \beta }$
$=\frac{1}{5\alpha \beta +2\left( \alpha ^{2} +\beta ^{2}\right)}$
$=\frac{1}{5\alpha \beta +2\left(( \alpha +\beta )^{2} -2\alpha \beta \right)}$
$=\frac{1}{5( -2) +2\left(( 3)^{2} -2( -2)\right)}$
$=\frac{1}{-10+2( 9+4)}$
$=\frac{1}{-10+26}$
$=\frac{1}{16}$
The quadratic polynomial having sum of the roots $S$ and product of the roots $P$ is $f(x)=k(x^2-(S)x+P)$, where $k$ is any non-zero real number.
Therefore,
$f(x)=k(x^2-(\frac{9}{16})x+\frac{1}{16})$
$f(x)=k(x^2-\frac{9}{16}x+\frac{1}{16})$
The required quadratic polynomial is $f(x)=k(x^2-\frac{9}{16}x+\frac{1}{16})$, where $k$ is any non-zero real number.
46 Views
