lpetrich
Contributor
Find
\( \displaystyle{ \sqrt{7 - \sqrt{7 + \sqrt{7 - \sqrt{7 + \dots} } } } } \)
This is x in
\( \displaystyle{ x = \sqrt{7 - \sqrt{7 + x} } } \)
Square it:
\( \displaystyle{ x^2 = 7 - \sqrt{7 + x} } \)
Subtract 7 and square again:
\( \displaystyle{ (x^2-7)^2 = 7 + x } \)
This expands into
\( x^4 - 14 x^2 - x + 42 \)
One can factor this polynomial by finding which gives zero out of everything that evenly divides 42, using both positive and negative signs. This gives x = 2 and x = -3. Thus
\( x^4 - 14 x^2 - x + 42 = (x - 2) (x + 3) (x^2 - x - 7) \)
The quadratic part has solutions
\( \displaystyle{ x = \frac12 \left( 1 \pm \sqrt{29} \right) } \)
The possible solutions to the original equation must have all square roots positive. This rules out the two negative solutions, so let us consider the remaining two. The one with sqrt(29) gives the wrong value, which means that that solution requires a negative sign for the inner square root. The remaining solution is x = 2:
\( \displaystyle{ \sqrt{7 - \sqrt{7 + 2} } = \sqrt{7 - \sqrt{9} } = \sqrt{7 - 3} = \sqrt{4} = 2 } \)
\( \displaystyle{ \sqrt{7 - \sqrt{7 + \sqrt{7 - \sqrt{7 + \dots} } } } } \)
This is x in
\( \displaystyle{ x = \sqrt{7 - \sqrt{7 + x} } } \)
Square it:
\( \displaystyle{ x^2 = 7 - \sqrt{7 + x} } \)
Subtract 7 and square again:
\( \displaystyle{ (x^2-7)^2 = 7 + x } \)
This expands into
\( x^4 - 14 x^2 - x + 42 \)
One can factor this polynomial by finding which gives zero out of everything that evenly divides 42, using both positive and negative signs. This gives x = 2 and x = -3. Thus
\( x^4 - 14 x^2 - x + 42 = (x - 2) (x + 3) (x^2 - x - 7) \)
The quadratic part has solutions
\( \displaystyle{ x = \frac12 \left( 1 \pm \sqrt{29} \right) } \)
The possible solutions to the original equation must have all square roots positive. This rules out the two negative solutions, so let us consider the remaining two. The one with sqrt(29) gives the wrong value, which means that that solution requires a negative sign for the inner square root. The remaining solution is x = 2:
\( \displaystyle{ \sqrt{7 - \sqrt{7 + 2} } = \sqrt{7 - \sqrt{9} } = \sqrt{7 - 3} = \sqrt{4} = 2 } \)