Answer:
x = 3/2 + sqrt(5)/2 or x = 3/2 - sqrt(5)/2
Step-by-step explanation:
Solve for x:
(x^3 - 4 x^2 + 4 x - 1)/(x - 1) = 0
Multiply both sides by x - 1:
x^3 - 4 x^2 + 4 x - 1 = 0
The left hand side factors into a product with two terms:
(x - 1) (x^2 - 3 x + 1) = 0
Split into two equations:
x - 1 = 0 or x^2 - 3 x + 1 = 0
Add 1 to both sides:
x = 1 or x^2 - 3 x + 1 = 0
Subtract 1 from both sides:
x = 1 or x^2 - 3 x = -1
Add 9/4 to both sides:
x = 1 or x^2 - 3 x + 9/4 = 5/4
Write the left hand side as a square:
x = 1 or (x - 3/2)^2 = 5/4
Take the square root of both sides:
x = 1 or x - 3/2 = sqrt(5)/2 or x - 3/2 = -sqrt(5)/2
Add 3/2 to both sides:
x = 1 or x = 3/2 + sqrt(5)/2 or x - 3/2 = -sqrt(5)/2
Add 3/2 to both sides:
x = 1 or x = 3/2 + sqrt(5)/2 or x = 3/2 - sqrt(5)/2
(x^3 - 4 x^2 + 4 x - 1)/(x - 1) ⇒ (-1 + 4 1 - 4 1^2 + 1^3)/(1 - 1) = (undefined):
So this solution is incorrect
(x^3 - 4 x^2 + 4 x - 1)/(x - 1) ⇒ (-1 + 4 (3/2 - sqrt(5)/2) - 4 (3/2 - sqrt(5)/2)^2 + (3/2 - sqrt(5)/2)^3)/((3/2 - sqrt(5)/2) - 1) = 0:
So this solution is correct
(x^3 - 4 x^2 + 4 x - 1)/(x - 1) ⇒ (-1 + 4 (sqrt(5)/2 + 3/2) - 4 (sqrt(5)/2 + 3/2)^2 + (sqrt(5)/2 + 3/2)^3)/((sqrt(5)/2 + 3/2) - 1) = 0:
So this solution is correct
The solutions are:
Answer: x = 3/2 + sqrt(5)/2 or x = 3/2 - sqrt(5)/2
