Question

If $$(x + 5)$$ is the geometric mean between $$(x + 2)$$ and $$(x + 9)$$, find the value of $$x$$.

Answer 1

Since $$(x + 5)$$ is the geometric mean (mean proportional) between $$(x + 2)$$ and $$(x + 9)$$:

$$(x + 5)^2 = (x + 2)(x + 9)$$

Expanding both sides:

$$x^2 + 10x + 25 = x^2 + 9x + 2x + 18$$

$$x^2 + 10x + 25 = x^2 + 11x + 18$$

Subtracting $$x^2$$ from both sides:

$$10x + 25 = 11x + 18$$

$$11x - 10x = 25 - 18$$

$$x = 7$$

Hence, the value of $$x$$ is 7.