# What is #sqrt(12+sqrt(12+sqrt(12+...)))# ?

##### 1 Answer

#### Explanation:

Suppose:

#color(blue)(c = sqrt(12+sqrt(12+sqrt(12+...))))#

Note that

Then:

#color(green)(sqrt(12+color(blue)(c)) = sqrt(12+color(blue)(sqrt(12+sqrt(12+sqrt(12+...)))))) = color(blue)(c)#

Squaring both ends, we find:

#12+c = c^2#

Note that squaring both sides of an equation results in an equation which must hold in order that the original equation holds, but is not necessarily sufficient. In this example, we will find a spurious negative solution for

Subtract

#0 = c^2-c-12 = (c-4)(c+3)#

So

We can discard the extraneous value

So the correct solution is