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 #3
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-1
Feb 8, 2016
 #1
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+10

Solve for x: -3-10 sqrt(x)+x^(3/2) = 0      Simplify and substitute y = sqrt(x): -3-10 sqrt(x)+x^(3/2) = -3-10 sqrt(x)+sqrt(x)^3 = y^3-10 y-3 = 0: y^3-10 y-3 = 0 The left hand side factors into a product with two terms: (y+3) (y^2-3 y-1) = 0 Split into two equations: y+3 = 0 or y^2-3 y-1 = 0 Subtract 3 from both sides: y = -3 or y^2-3 y-1 = 0 Substitute back for y = sqrt(x): sqrt(x) = -3 or y^2-3 y-1 = 0 Raise both sides to the power of two: x = 9 or y^2-3 y-1 = 0 Add 1 to both sides: x = 9 or y^2-3 y = 1 Add 9/4 to both sides: x = 9 or y^2-3 y+9/4 = 13/4 Write the left hand side as a square: x = 9 or (y-3/2)^2 = 13/4 Take the square root of both sides: x = 9 or y-3/2 = sqrt(13)/2 or y-3/2 = -sqrt(13)/2 Add 3/2 to both sides: x = 9 or y = 3/2+sqrt(13)/2 or y-3/2 = -sqrt(13)/2 Substitute back for y = sqrt(x): x = 9 or sqrt(x) = 3/2+sqrt(13)/2 or y-3/2 = -sqrt(13)/2 Raise both sides to the power of two: x = 9 or x = (3/2+sqrt(13)/2)^2 or y-3/2 = -sqrt(13)/2 Add 3/2 to both sides: x = 9 or x = (3/2+sqrt(13)/2)^2 or y = 3/2-sqrt(13)/2 Substitute back for y = sqrt(x): x = 9 or x = (3/2+sqrt(13)/2)^2 or sqrt(x) = 3/2-sqrt(13)/2 Raise both sides to the power of two: x = 9 or x = (3/2+sqrt(13)/2)^2 or x = (3/2-sqrt(13)/2)^2 -3-10 sqrt(x)+x^(3/2) => -3-10 sqrt(9)+9^(3/2) = -6: So this solution is incorrect -3-10 sqrt(x)+x^(3/2) => -3-10 sqrt((3/2-sqrt(13)/2)^2)+((3/2-sqrt(13)/2)^2)^(3/2) = -6: So this solution is incorrect -3-10 sqrt(x)+x^(3/2) => -3-10 sqrt((3/2+sqrt(13)/2)^2)+((3/2+sqrt(13)/2)^2)^(3/2) = 0: So this solution is correct The solution is: Answer: | | x = (3/2+sqrt(13)/2)^2 =~10.908......

 

 x+1/x=?=10.908 + 1/10.908=~11

Feb 8, 2016
 #2
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Feb 8, 2016

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