Question

For what values of $$x$$, the numbers $$(x + 9), (x - 6)$$ and 4 are in G.P.?

Answer 1

For three numbers to be in G.P., the square of the middle number equals the product of the first and the third.

$$(x - 6)^2 = (x + 9) \times 4$$

$$x^2 - 12x + 36 = 4x + 36$$

$$x^2 - 16x = 0 \Rightarrow x(x - 16) = 0$$

$$x = 0 \text{ or } x = 16$$

Check:

x = 0: the numbers are 9, –6, 4 and $$\frac{-6}{9} = \frac{4}{-6} = -\frac{2}{3}$$ ✓

x = 16: the numbers are 25, 10, 4 and $$\frac{10}{25} = \frac{4}{10} = \frac{2}{5}$$ ✓

So, x = 0 or x = 16.