3/4=1/2x^(1/2)-1/2x^(-1/2)

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Solution for 3/4=1/2x^(1/2)-1/2x^(-1/2) equation: 


D( x )

x < 0

x = 0

x < 0

x = 0

x = 0

x in (0:+oo)

3/4 = (1/2)*x^(1/2)-((1/2)*x^(-1/2)) // - (1/2)*x^(1/2)-((1/2)*x^(-1/2))

(1/2)*x^(-1/2)-((1/2)*x^(1/2))+3/4 = 0

(-1/2)*x^(1/2)+(1/2)*x^(-1/2)+3/4 = 0

1/2*x^(-1/2)-1/2*x^(1/2)+3/4 = 0

t_1 = x^(1/2)

1/2*t_1^-1-1/2*t_1^1+3/4 = 0

1/2*t_1^-1-1/2*t_1^1+3/4*t_1^0 = 0

(3/4*t_1^1-1/2*t_1^2+1/2*t_1^0)/(t_1^1) = 0 // * t_1^2

t_1^1*(3/4*t_1^1-1/2*t_1^2+1/2*t_1^0) = 0

t_1^1

(-1/2)*t_1^2+(3/4)*t_1+1/2 = 0

(-1/2)*t_1^2+(3/4)*t_1+1/2 = 0

DELTA = (3/4)^2-(4*(-1/2)*(1/2))

DELTA = 25/16

DELTA > 0

t_1 = ((25/16)^(1/2)-(3/4))/(2*(-1/2)) or t_1 = (-(3/4)-(25/16)^(1/2))/(2*(-1/2))

t_1 = -1/2 or t_1 = 2

t_1 in { -1/2, 2} 

t_1 = -1/2

x^(1/2)+1/2 = 0

1*x^(1/2) = -1/2 // : 1

x^(1/2) = -1/2

( -1/2 < 0 i 1/2 in (0:1) ) => x należy do O

t_1 = 2

x^(1/2)-2 = 0

1*x^(1/2) = 2 // : 1

x^(1/2) = 2

x^(1/2) = 2 // ^ 2

x = 4

x = 4

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