Question

If `alpha` and `beta` are the zeros of the quadratic polynomial `f(x)=2x^2-5x+7` , find a polynomial whose zeros are `2alpha+3beta` and `3alpha+2beta` .

Answer 1

Quadratic equation is of the form `ax^2+bx+c`
now sum of roots=`(-b)/a` and product of roots=`c/a`
here the eqn is `2x^2-5x+7` and `alpha ` and `beta` are its roots.
sum of roots=`alpha + beta`=`5/2` and product of roots=`alpha xx beta`=`7/2`
now we have to form an equation whose roots are `2alpha +3beta` and `3alpha + 2beta`.
equation will be of the form `x^2-sx+p`......(1) where s and p represent sum and product of roots.
s=`5(alpha+beta)`=`25/2` and p=`6((alpha)^2+(beta)^2)+13alphabeta)`=`6(alpha+beta)^2+alphabeta`=`6/4 xx 25 +7/2`=41.
substituting s and p in (1) we get eqn=`x^2-25/2x+41=0`.