Cross between alg and geom

Algebra Level 3

The arithmetic mean of two variables $a,b$ is 4, and the quadratic mean is 5. If the geometric mean can be represented as $\sqrt{x}$ , find $x$ .

The answer is 7.

2 solutions

Trevor Arashiro
Jul 25, 2014

See this link

This image really helps understand the logic behind this solution.

If the Arithmetic mean is 4 and the quadratic mean is 5, than by Pythagorean theorem, the base of the right triangle will be three; with one leg representing AM and the hypotenuse representing QM. Thus the other right triangle with 3 as its base, the other leg being the GM, and the hypotenuse representing the AM=4. The Pythagorean theorem tells us that the GM $=\sqrt{7}$ .

Btw, what's going on with the LaTeX. Ever since brilliant updated the site some LaTeX codes aren't working.

nice solution trevor ! thumbs up .. would certainly make life easier , very new technique .. i solved equations

Devank Yadav - 6 years, 10 months ago

Hey Trevor, I think what's going on is that you have to wrap $\LaTeX$ commands in \) and \( .

Josh Silverman Staff - 6 years, 10 months ago

Ooo, actually, I forgot the \before the sqrt, you need two of those.

Trevor Arashiro - 6 years, 10 months ago

I did, $sqrt{x}$

Trevor Arashiro - 6 years, 10 months ago

And that's what happens

Trevor Arashiro - 6 years, 10 months ago

Excelent solution!

Carlos David Nexans - 6 years, 10 months ago

Muzzammal Alfath
Jul 25, 2014

AM _ (a+b)/2=4.......a+b=8. QM_ ((a^2+b^2)/2)^(1/2)=5......a^2+b^2=50. We get ab=7 . Then GM_(ab)^(1/2)= 7^(1/2). Then the value of x=7

Cross between alg and geom

The answer is 7.

2 solutions

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