Question 1135942
<pre>
{{{sqrt(x - 127) + sqrt(k - x) = 13}}}

Isolate a square root term:

{{{sqrt(x - 127) = 13 - sqrt(k - x)}}}

Square both sides:

{{{x - 127 = (13 - sqrt(k - x))^2}}}

{{{x - 127 = (13 - sqrt(k - x))(13 - sqrt(k - x))}}}

{{{x - 127 = 169-26sqrt(k-x)+(k-x)}}}

Add 127 to both sides

{{{x = 296-26sqrt(k-x)+k-x}}}

Add x to both sides:

{{{2x = 296-26sqrt(k-x)+k}}}

Isolate the square root term

{{{26sqrt(k-x)= 296-2x+k}}}

Square both sides:

{{{676(k-x)= (296-2x+k)^2}}}

{{{676k-676x = 87616 + 4x^2 + k^2 - 1184x + 592k - 4kx}}}

{{{-4x^2-676x + 1184x + 4kx - 87616 - k^2 - 592k + 676k = 0}}}

{{{-4x^2+508x + 4kx - 87616 - k^2 + 84k = 0}}}

Change all signs

{{{4x^2-508x - 4kx + 87616 + k^2 - 84k = 0}}}

{{{4x^2+(-508 - 4k)x + k^2 - 84k + 87616 = 0}}}

{{{4x^2+(-508 - 4k)x + (k^2 - 84k + 87616) = 0}}}

This quadratic will have a real solution if and only if
its discriminant is not negative:

Discriminant = B²-4AC =

{{{(-508 - 4k)^2-4(4)(k^2 - 84k + 87616)}}}

{{{(-508 - 4k)(-508-4k)-16(k^2 - 84k + 87616)}}}

{{{258064 + 4064k + 16k^2 -16k^2 + 1344k-1401856}}}

{{{5408k-1143792}}}

{{{2704(2k - 423)}}}

This discriminant will be nonnegative if

{{{2k-423>=0}}}

{{{2k>=423}}}

{{{k>=211.5}}}

So the answer is the next integer after 211.5 which is 212.

Edwin</pre>