Solve for x over the real numbers: x^3-2 x^2+1 = 0 The left hand side factors into a product with two terms: (x-1) (x^2-x-1) = 0 Split into two equations: x-1 = 0 or x^2-x-1 = 0 Add 1 to both sides: x = 1 or x^2-x-1 = 0 Add 1 to both sides: x = 1 or x^2-x = 1 Add 1/4 to both sides: x = 1 or x^2-x+1/4 = 5/4 Write the left hand side as a square: x = 1 or (x-1/2)^2 = 5/4 Take the square root of both sides: x = 1 or x-1/2 = sqrt(5)/2 or x-1/2 = -sqrt(5)/2 Add 1/2 to both sides: x = 1 or x = 1/2+sqrt(5)/2 or x-1/2 = -sqrt(5)/2 Add 1/2 to both sides: Answer: | | x = 1 or x = 1/2+sqrt(5)/2 or x = 1/2-sqrt(5)/2
Posted on: Mon, 10 Nov 2014 01:48:14 +0000
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