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 :
[tex]GM= \sqrt[n]{a_1a_2a_3...a_n} [/tex]
So:
[tex] \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)[/tex]
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,
[tex] \sqrt{2(2)(19*2-2)} =7(2)-2 \\ \sqrt{4(36)}=12 \\ 12=12[/tex]
This is true therefore x can be 2.
When x = 2/11
[tex] \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} [/tex]
This is not true therefore x cannot be 2/11.
The only possible value of x is then 2.
