Question

how to solve \sqrt{t - 14} - \sqrt{t+96} = 9

Answer 1

t+96 will always be larger than t-14
so it looks like there will be no solution.

but we can proceed as follows:
square both sides
\[ \sqrt{t - 14} - \sqrt{t+96} = 9\\
t-14 + t+96 - 2 \sqrt{(t-14)(t+96)}=81\\
2t + (96-14-81) = 2 \sqrt{(t-14)(t+96)} \\
2t+1 = 2 \sqrt{(t-14)(t+96)}
\]

you can now square both sides again
\[ 4t^2 +4t+1 = 4(t-14)(t+96) \\
4t^2 +4t+1 = 4 ( t^2 +82t - 1344)\\
\cancel{4t^2} +4t+1 = \cancel{4t^2} + 328t - 5376 \\
5377 = 324 t\\
t= \frac{5377}{324} \approx 16.59567901
\]

if we try that solution we get the left hand side = -9
so it is an extraneous solution... and there is no solution