$(x - 4)$ is geometric mean of $(x - 5)$ and $(x - 2)$ , find the value of $x$ .

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Solution

Geometric mean of two numbers, $a$ and $b$ is $\sqrt{a b}$
So, geometric mean of $(x - 5), (x - 2)$ is $\sqrt{(x - 5) \times (x - 2)} = (x - 4)$
Squaring both sides,
$x^{2} - 5 x - 2 x + 10 = x^{2} - 8 x + 16$
$- 7 x + 8 x = 16 - 10$
$x = 6$

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