CBSE Class 10 Answered

Let alpha and beta are the zeroes of a quadratic polynomial 2x²-5x-6 then form a quadratic polynomial whose zeroes are alpha+beta and alpha×beta

Asked by geetayadav18081983 | 15 Jun, 2022, 01:19: PM

Expert Answer

Given quadratic equation is

f(x) = 2x² -5x - 6

roots of above quadratic equation are

begin mathsize 14px style fraction numerator 5 plus square root of 25 minus 4 left parenthesis 2 right parenthesis left parenthesis negative 6 right parenthesis end root over denominator 2 cross times 2 end fraction end style

and

begin mathsize 14px style fraction numerator 5 minus square root of 25 minus 4 left parenthesis 2 right parenthesis left parenthesis negative 6 right parenthesis end root over denominator 2 cross times 2 end fraction end style

roots of above quadratic equation are

begin mathsize 14px style fraction numerator 5 plus square root of 73 over denominator 4 end fraction end style

and

begin mathsize 14px style fraction numerator 5 minus square root of 73 over denominator 4 end fraction end style

Hence

begin mathsize 14px style alpha space equals space fraction numerator 5 plus square root of 73 over denominator 4 end fraction end style

and

begin mathsize 14px style beta space equals space fraction numerator 5 minus square root of 73 over denominator 4 end fraction end style

If roots of a quadratic equition are (α + β ) and ( α β ) , then quadratic equation is

g(x) = x² - (α + β + α β ) x + (α + β ) ( α β )

begin mathsize 14px style alpha space plus beta equals space fraction numerator 5 plus square root of 73 over denominator 4 end fraction plus fraction numerator 5 minus square root of 73 over denominator 4 end fraction space equals space 5 over 2 end style

begin mathsize 14px style alpha space beta space equals space open parentheses space fraction numerator 5 plus square root of 73 over denominator 4 end fraction space close parentheses space space stretchy left parenthesis space fraction numerator 5 minus square root of 73 over denominator 4 end fraction space stretchy right parenthesis space equals space 1 over 16 open parentheses 25 plus 73 minus 10 square root of 73 close parentheses space equals 49 over 8 minus 5 over 8 square root of 73 end style

Hence quadratic equation is

begin mathsize 14px style g left parenthesis x right parenthesis space equals space x squared space minus space open parentheses 5 over 2 space plus space 49 over 8 space minus space 5 over 8 square root of 73 close parentheses x space plus 5 over 16 open parentheses 49 minus 5 square root of 73 close parentheses end style

begin mathsize 14px style g left parenthesis x right parenthesis space equals space x squared space minus space open parentheses space 69 over 8 space minus space 5 over 8 square root of 73 close parentheses x space plus 5 over 16 open parentheses 49 minus 5 square root of 73 close parentheses end style

begin mathsize 14px style g left parenthesis x right parenthesis space equals space 16 space x squared space minus space open parentheses space 138 space minus space 10 square root of 73 close parentheses x space plus open parentheses 245 minus 25 square root of 73 close parentheses end style

Answered by Thiyagarajan K | 15 Jun, 2022, 02:43: PM

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