Question #51c56

#sqrtx div sqrt(1-x)+sqrt(1-x)=5÷2#

#rArr sqrtx/sqrt(1 - x) + sqrt (1 - x) = 5/2#

Taking L.C.M

#rArr (sqrtx + 1 - x)/sqrt(1 - x) = 5/2#

Cross Multiply

#rArr 2 (sqrtx + 1 - x) = 5 sqrt(1 - x)#

#rArr 2sqrtx + 2 - 2x = 5 sqrt(1 - x)#

Collect Like Terms

#rArr 2sqrtx - 5 sqrt(1 - x) = 2x + 2#

Square both sides

#rArr (2sqrtx - 5 sqrt(1 - x))^2 = (2x + 2)^2#

#rArr (2sqrtx - 5 sqrt(1 - x)) (2sqrtx - 5 sqrt(1 - x)) = (2x + 2) (2x + 2)#

#rArr 4(x) - 20sqrt(1 - x) + 25(1 - x) = 4x^2 + 8x + 4#

#rArr 4(x) - 20sqrt(1 - x) + 25 - 25x = 4x^2 + 8x + 4#

#rArr- 20sqrt(1 - x) + 25 - 21x = 4x^2 + 8x + 4#

#rArr- 20sqrt(1 - x) = 4x^2 + 8x + 4 +21x - 25#

#rArr- 20sqrt(1 - x) = 4x^2 + 29x - 21#

Square both sides

#rArr (- 20sqrt(1 - x))^2 = (4x^2 + 29x - 21)^2#

#rArr 400 (1 - x) = (4x^2 + 29x - 21)^2#

#rArr 400 - 400x = (4x^2 + 29x - 21) (4x^2 + 29x - 21)#

#rArr 400 - 400x = 16x^4 + 232x^3 + 673x^2 - 1218x + 441#

Collect like terms

#rArr 16x^4 + 232x^3 + 673x^2 - 1218x + 400x + 441 - 400 = 0#

#rArr 16x^4 + 232x^3 + 673x^2 - 818x + 41 = 0#

Solve the polynomial above..

That's how far i could get, But in my own point of view, i strongly doubt the Authenticity of the question..

#sqrtx/sqrt(1-x)+sqrt(1-x)=2.5# Calling #y = 1-x# we have

#sqrt(1-y)+y=2.5 sqrty# or

#1-y=(2.5 sqrty-y)^2# or

#1-y=2.5^2y-5y sqrty+y^2# and now making #xi=sqrty# we have

#xi^4-5xi^3+(1+2.5^2)xi^2-1=0# This polynomial has two real roots

#xi = {-0.332869,0.43602}# giving

#y = {0.110802,0.190114}# and consequently

#x = {0.889198,0.809886}#

but substituting into the original equation only #x = 0.809886# is feasible, so the solution is

# x = 0.809886#