PROBLEM: Given 
and 
are two quadratic polynomials with rational coefficients.
Suppose 
and 
have a common irrational solution.
Prove that 
for all 
where 
is a rational number.
SOLUTION: Suppose the common irrational root of (\ f(x)) and (\ g(x)) be (\sqrt{a}+b).
Then by properties of irrational roots we can say that the other root of both of them will be (\sqrt{a}-b).
so we can write (\ f(x)=\lambda(x-\sqrt{a}-b)(x-\sqrt{a}+b)) and (\ g(x)=\mu(x-\sqrt{a}-b)(x-\sqrt{a}+b))
so (\frac{g(x)}{f(x)}=\frac{\mu}{\lambda})
therefore,$$\ g(x)=f(x)\frac{\mu}{\lambda}=rf(x)$$.
Theorem:In an equation with real coefficients irrational roots occurs in conjugate pairs.
