**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.

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