if α and β are the roots ofax²+bx+c=0, then find roots ofa(x-2)²-b(x-2)(x-3)+c(x-3)²=0,

Put alpha in ax^2+bx+c=0,it becomes a(alpha)^2+b(alpha)+c=0[as alpha is given as root of this equation]. Eq1

From a(x-2)^2 + b(x-2)(x-3) +c(x-3)^2=0

Take (x-3)^2 as common,it becomes

a{(x-2)/(x-3)}^2+b{(x-2)/(x-3)} +c=0. Eq2

Now compare the coefficient of a in Eq1 and Eq2

We get aplha=x-2/x-3

From here x=(3alpha-2)/alpha-1

 

 

Dear student 

Dear student 
ax^2 + bx(x-1) +c(x-1)^2 = 0
ax^2 + bx^2 +bx + c^2 + c - 2cx =0
x^2 (a-b+c) -x(2c -b) + c = 0 
p+q = (2c -b)/a-b+c
p.q = (c/a)/1-b/a+c/a
p.q = alpha.Beta /(1-alpha + beta +alpha.beta)
p+q = (alpha/alpha+1) + (beta/beta + 1)
p.q = (alpha / alpha +1 ). (Beta/beta + 1)