CBSE Class 10 Answered

ifalfa+beta are roots of f(x)=x2_3x_2find quadratic polynomial whose roots are 1/2alfa+betaand1/2beta+alfa

Asked by cutepawan87 | 05 Sep, 2015, 02:50: PM

Expert Answer

$C o n s i d e r space t h e space e q u a t i o n comma space f open parentheses x close parentheses equals x squared minus 3 x minus 2 L e t space alpha space a n d space beta space a r e space t h e space r o o t s space o f space t h e space q u a d r a t i c space p o l y n o m i a l. T h u s comma space alpha plus beta equals negative open parentheses negative 3 close parentheses space a n d space alpha beta equals negative 2 rightwards double arrow space alpha plus beta equals 3 space a n d space alpha beta equals negative 2 C o n s i d e r space t h e space g i v e n space e x p r e s s i o n comma space fraction numerator 1 over denominator 2 alpha end fraction plus beta plus fraction numerator 1 over denominator 2 beta end fraction plus alpha fraction numerator 1 over denominator 2 alpha end fraction plus beta plus fraction numerator 1 over denominator 2 beta end fraction plus alpha equals fraction numerator 2 beta plus 4 alpha beta squared plus 2 alpha plus 4 alpha squared beta over denominator 4 space alpha beta end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 open parentheses alpha plus beta close parentheses plus 4 alpha beta open parentheses alpha plus beta close parentheses over denominator 4 space alpha beta end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 cross times 3 plus 4 cross times open parentheses negative 2 close parentheses cross times 3 over denominator 4 space cross times open parentheses negative 2 close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 6 minus 24 over denominator negative 8 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator negative 18 over denominator negative 8 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 9 over 4 C o n s i d e r space t h e space g i v e n space e x p r e s s i o n comma open parentheses fraction numerator 1 over denominator 2 alpha end fraction plus beta close parentheses cross times open parentheses fraction numerator 1 over denominator 2 beta end fraction plus alpha close parentheses equals 1 plus fraction numerator 1 over denominator 4 alpha beta end fraction plus alpha beta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 plus fraction numerator 1 over denominator 4 cross times open parentheses negative 2 close parentheses end fraction plus open parentheses negative 2 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 minus 1 over 8 minus 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals negative 9 over 8 T h u s comma space t h e space r e q u i r e d space e q u a t i o n space i s x squared minus open parentheses s u m space o f space t h e space r o o t s close parentheses x plus open parentheses p r o d u c t space o f space t h e space r o o t s close parentheses T h u s comma space r e q u i r e d space e q u a t i o n space i s space x squared minus open parentheses 9 over 4 close parentheses x plus open parentheses negative 9 over 8 close parentheses rightwards double arrow 8 x squared minus 8 cross times open parentheses 9 over 4 close parentheses x plus 8 cross times open parentheses negative 9 over 8 close parentheses rightwards double arrow 8 x squared minus 18 x minus 9$