Chapter 11.10, Problem 10P

To determine

To solve: Two consecutive positive even integers whose geometric mean is $4\sqrt{39}$ .

Answer to Problem 10P

The integers are 24 and 26.

Explanation of Solution

Given:

The geometric mean of two positive numbers is the positive square root of their product.

Calculation:

Let the consecutive positive even integers be m and $m+2$ .

Then the geometric mean is given by $\sqrt{m\left(m+2\right)}$ which is equal to $4\sqrt{39}$ . So,

  $\begin{array}{c}\sqrt{m\left(m+2\right)}=4\sqrt{39}\\ m\left(m+2\right)=\left(16\right)\left(39\right)\\ m^2+2m=624\\ m^2+2m-624=0\end{array}$

Factorize the equation by using the splitting the middle term method.

  $\begin{array}{c}{m}^2+2m-624=0\\ m^2+26m-24m-624=0\\ m\left(m+26\right)-24\left(m+26\right)=0\\ \left(m+26\right)\left(m-24\right)=0\end{array}$

By the zero product property, either $m+26=0$ or $m-24=0$ which implies that either $m=-26$ or $m=24$ .

Since m is a positive integer, the only possible value of m is 24. So, the integers are 24 and 26.