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The geometric mean is similar to the arithmetic mean (or average). The geometric mean of n terms is equal to the nth root of the n terms or :
GM= \sqrt[n]{a_1a_2a_3...a_n}

 \sqrt{2x(19x-2)} =7x-2\\ \sqrt{38x^2-4x} =7x-2 \\ 38x^2-4x=49x^2-28x+4 \\ 0=11x^2-24x+4 \\ 0=(11x-2)(x-2)

So x can be equal to 2 or 2/11. We check if these are extraneous roots (meaning they do not work).

When x = 2,
 \sqrt{2(2)(19*2-2)} =7(2)-2 \\  \sqrt{4(36)}=12 \\ 12=12
This is true therefore x can be 2.

When x = 2/11
 \sqrt{2( \frac{2}{11})(19* \frac{2}{11} -2 )}=7( \frac{2}{11}  )-2 \\  \sqrt{ \frac{4}{11}( \frac{16}{11})  } = -\frac{8}{11}  \\  \frac{8}{11} = -\frac{8}{11}
This is not true therefore x cannot be 2/11.

The only possible value of x is then 2.