Anish Singhal
Last Activity: 6 Years ago
Letx1 be the irrational root that is shared byp and q. Then, by the Irrational Conjugate Roots Theorem, the irrational conjugate ofx1x1must also be a root of bothp andq.
Explicitly, for any rationala,band irrational √c, ifx1=a+brootc
is a root of p, then
x2=a−brootc
must also be a root. The same reasoning implies that x^2is a root of q.Since pand qare quadratics whose two roots are the same, they must be proportional, up to a constant. Since we have integer coefficients, the constant of proportionality rmust be rational, so we are done.