Question

the geometric mean of two numbers is 2\sqrt{5} one of the numbers is 6. find the other number
*******metal ************

Answer 1

Okay...well to find the geometric mean...you take the two numbers.....place them under a square root sign (since it is 2 numbers)...and multiply them.. so

\[\sqrt{6 \times x} = \frac{ 2 }{ \sqrt{5} }\]

So like the last time....lets square both sides to get rid of the square root signs

\[\sqrt{6 \times x}^2 = (\frac{ 2 }{ \sqrt{5} })^2\]

becomes

\[6x = \frac{ 4 }{ 5 }\]

Lets multiply both sides by 5

\[30x = 4\]

And divide both sides by 30 to solve for 'x'

\[x = \frac{ 4 }{ 30 }\]

and simplify

\[x = \frac{ 2 }{ 15 }\]

Now lets check by putting it in the original equation

\[\sqrt{6 \times \frac{ 2 }{ 15 }} = \frac{ 2 }{ \sqrt{5} }\]
\[\sqrt{\frac{ 12 }{ 15 }} = \frac{ 2 }{ \sqrt{5} }\]

this can be written as

\[\frac{ \sqrt{4}\sqrt{3} }{ \sqrt{5}\sqrt{3} } = \frac{ 2\sqrt{3} }{ \sqrt{5}\sqrt{3} } = \frac{ 2 }{ \sqrt{5} }\]

so yes it is correct....the other number is 2/15