Question

if a and b are the zeroes of the quadratic equation f(x)=
${x }^{2} - 3x - 2$
,find a quadratic polynomial whose zeroes are 1/(2a+b) and 1/(2b+a) .

Answer 1

f (x) = x^2 - 3x -2
f {1/(2a+b)} ={1/(2a+b)}^2 -3 {1/(2a+b)} -2

taking L.C.M
= {1- 3 (2a+b)-2 (2a+b)^2}/(2a+b)^2
= {1-6a-3b-2 (4a^2+b^2+4ab)}/(2a+b)^2
= {1-6a-3b-8a^2-2b^2-8ab}/(2a +b)^2
= -(8a^2+2b^2+8ab+6a+3b-1)/(2a+b)^2
similarly solve for 1/(2b+a)

Answer 2

f (x) = x^2 - 3x -2
f {1/(2a+b)} ={1/(2a+b)}^2 -3 {1/(2a+b)} -2

taking L.C.M
= {1- 3 (2a+b)-2 (2a+b)^2}/(2a+b)^2
= {1-6a-3b-2 (4a^2+b^2+4ab)}/(2a+b)^2
= {1-6a-3b-8a^2-2b^2-8ab}/(2a +b)^2
= -(8a^2+2b^2+8ab+6a+3b-1)/(2a+b)^2
similarly solve for 1/(2b+a)