**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.

Google