Find $m$, so that the equation $3{ x }^{ 2 }-2mx-4=0$ and ${ x }^{ 2 }-4mx+2=0$ may have a common root. Can the equations have a common non-real root?

Let the common root be a, so it satisfies both 3a^2 - 2ma - 4 = 0 and a^2 - 4ma + 2 = 0. Using the method of cross-multiplication for common roots, we get a^2 / (-4 - 8m^2) = a / (-4 - 12) = 1 / (-12m + 2m). This implies a^2 = (-4 - 8m^2) / (-16) and a = (-16) / (-10m) = 8 / 5m. Equating a and a^2 gives (8 / 5m)^2 = (4 + 8m^2) / 16, which simplifies to 64 / 25m^2 = (1 + 2m^2) / 4. Solving the resulting equation 50m^4 - 256 = 25m^2 leads to no real solutions for m. Because the coefficients are real, any non-real roots must occur in conjugate pairs, meaning they cannot be uniquely shared unless the equations are identical, which they are not.

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