If alpha and beta are the roots of the equation x^2+5x+3=0,find the value of (1/beta^3)-(1/alpha).

Asked by nitinkumarsingh.singh.75 | 23rd Jul, 2015, 10:43: AM

Expert Answer:

G i v e n space e q u a t i o n space i s space x squared plus 5 x plus 3 equals 0 L e t space alpha space a n d space beta a r e space t h e space r o o t s space o f space t h e space e q u a t i o n. therefore alpha plus beta equals fraction numerator negative 5 over denominator 1 end fraction equals negative 5 a n d alpha beta equals 3 over 1 equals 3 W e space n e e d space t o space f i n d space t h e space v a l u e space o f space 1 over beta cubed minus 1 over alpha cubed 1 over beta cubed minus 1 over alpha cubed equals fraction numerator alpha cubed minus beta cubed over denominator alpha cubed beta cubed end fraction W e space k n o w space t h a t comma space alpha cubed minus beta cubed space equals space open parentheses alpha minus beta close parentheses open parentheses alpha squared plus alpha beta plus beta squared close parentheses T h u s comma space alpha squared plus alpha beta plus beta squared equals alpha squared plus beta squared plus alpha beta space space space space space space space space space space space space space space space space space space space equals open parentheses alpha plus beta close parentheses squared minus 2 alpha beta plus alpha beta space space space space space space space space space space space space space space space space space space space equals open parentheses alpha plus beta close parentheses squared minus alpha beta space space space space space space space space space space space space space space space space space space space equals open parentheses negative 5 close parentheses squared minus 3 space space space space space space space space space space space space space space space space space space space equals 25 minus 3 space space space space space space space space space space space space space space space space space space space equals 22.... left parenthesis 1 right parenthesis open parentheses alpha minus beta close parentheses squared equals open parentheses alpha plus beta close parentheses squared minus 4 alpha beta space space space space space space space space space space space space equals open parentheses negative 5 close parentheses squared minus 4 cross times 3 space space space space space space space space space space space space equals 25 minus 12 space space space space space space space space space space space space equals 13 rightwards double arrow open parentheses alpha minus beta close parentheses equals square root of 13 T h e r e f o r e comma. alpha cubed minus beta cubed space equals space square root of 13 cross times 22 T h u s comma 1 over beta cubed minus 1 over alpha cubed equals fraction numerator alpha cubed minus beta cubed over denominator alpha cubed beta cubed end fraction space space space space space space space space space space space space space space space space equals fraction numerator 22 square root of 13 over denominator 3 cubed end fraction space space space space space space space space space space space space space space space space equals fraction numerator 22 square root of 13 over denominator 27 end fraction

Answered by Vimala Ramamurthy | 23rd Jul, 2015, 06:11: PM

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