Question

If it is given that $\left( x-4 \right)$ is a geometric mean of $\left( x-5 \right)$ and $\left( x-2 \right)$, then find the value of $x$.

Answer 1

In this problem we need to calculate the value of $x$ where the geometric mean of $\left( x-5 \right)$ and $\left( x-2 \right)$ is $\left( x-4 \right)$. We know that the geometric mean of the numbers $a$ and $b$ is $\sqrt{ab}$. So, we will first calculate the geometric mean of $\left( x-5 \right)$ and $\left( x-2 \right)$ by using the above formula. After that we will equate the calculated value of the geometric mean to the given geometric mean which is $\left( x-4 \right)$ and simplify the equation to get the required result.

Complete step by step answer:
Given that, the geometric mean of $\left( x-5 \right)$ and $\left( x-2 \right)$ is $\left( x-4 \right)$.
We know that the geometric mean of the numbers $a$ and $b$ is $\sqrt{ab}$. From this formula the geometric mean of the numbers $\left( x-5 \right)$ and $\left( x-2 \right)$ is given by
$m=\sqrt{\left( x-5 \right)\left( x-2 \right)}$
But in the problem, we have the geometric mean of the numbers as $\left( x-5 \right)$. So, equating the both the values, then we will get
$\left( x-4 \right)=\sqrt{\left( x-5 \right)\left( x-2 \right)}$
Squaring on both sides of the above equation, then we will have
${{\left( x-4 \right)}^{2}}=\left( x-5 \right)\left( x-2 \right)$
Using the formulas ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$, $\left( x-a \right)\left( x-b \right)={{x}^{2}}-\left( a+b \right)x+ab$in the above equation, then we will get
${{x}^{2}}-2\left( x \right)\left( 4 \right)+{{4}^{2}}={{x}^{2}}-\left( 5+2 \right)x+\left( 5 \right)\left( 2 \right)$
Simplifying the above equation by cancelling the common term ${{x}^{2}}$ which is on both sides of the equation, then we will get
$-8x+16=-7x+10$
Rearrange the terms in the above equation, so that all the variables at one side and constants at one side, then we will get
$\begin{aligned} & 8x-7x=16-10 \\\ & \therefore x=6 \\\ \end{aligned}$

Note: In this problem we don’t have calculated the geometric mean by simplifying the equation $m=\sqrt{\left( x-5 \right)\left( x-2 \right)}$ which takes too much of time and gives some complexity in further steps. So, use simple mathematical operations and solve the problem in an easiest way.