Question

If x + 3 is the geometric mean between x + 1 and x + 6 then find x.

Answer 1

We know that the geometric mean of two numbers a and b is $G=\pm \sqrt{ab}$.
We have:
G = x + 3, a = x + 1 and b = x + 6
Thus, we get:
$$\begin{gathered} x+3=\pm \sqrt{(x+1)(x+6)} \\ \mathrm{On}\mathrm{squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}: \\ (x+3{)}^2=(x+1\left)\right(x+6) \\ \Rightarrow x^2+6x+9=x^2+7x+6 \\ \Rightarrow 6x-7x=6-9 \\ \Rightarrow -x=-3 \\ \Rightarrow x=3 \end{gathered}$$