Question #51c56

2 Answers
Aug 17, 2017

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

Explanation:

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..

Aug 17, 2017

See below.

Explanation:

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