Here's my best attempt :
0 = w ( L^2x^2 - x^4 - 7L^4 )
____ ______ ___ _____
EI6L 2 4 60
Multiply both sides by EI6L / w and the lead term "disappears"
0 = L^2 x^2 - x^4 - 7L^4
______ ____ _____
2 4 60
Multiply through by - 60
15x^4 - 30L^2x^2 + 7L^4 = 0
15x^4 - 30L^2x^2 = - 7L^4 factor out 15
15 ( x^4 - 2L^2x^2) = - 7L^4 divide both sides by 15
x^4 - 2L^2x^2 = - 7L^4 / 15
Take 1/2 of 2L^2 = L^2
Square this = L^4
Add to both sides
x^4 - 2L^2 x^2 + L^4 = -7L^4 / 15 + L^4
Factor the left side and simplify
( x^2 - L^2)^2 = 8L^4 / 15
Take both roots and simplify
x^2 - L^2 = ±√ [ (8L^4 / 15) ]
x^2 - L^2 = ±2L^2√ [ 2/15 ]
Add L^2 to both sides
x^2 = L^2 ± 2L^2√ [ 2/15 ]
Take both roots again
x = ±√ ( L^2 ± 2L^2√ [ 2/15 ] ) = ± L√ ( 1 ± 2√ [ 2/15 ] )
