z^2-1z=z+1

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Solution for z^2-1z=z+1 equation:


Simplifying
z2 + -1z = z + 1

Reorder the terms:
-1z + z2 = z + 1

Reorder the terms:
-1z + z2 = 1 + z

Solving
-1z + z2 = 1 + z

Solving for variable 'z'.

Reorder the terms:
-1 + -1z + -1z + z2 = 1 + z + -1 + -1z

Combine like terms: -1z + -1z = -2z
-1 + -2z + z2 = 1 + z + -1 + -1z

Reorder the terms:
-1 + -2z + z2 = 1 + -1 + z + -1z

Combine like terms: 1 + -1 = 0
-1 + -2z + z2 = 0 + z + -1z
-1 + -2z + z2 = z + -1z

Combine like terms: z + -1z = 0
-1 + -2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -2z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2z + z2 = 0 + 1
-2z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2z + z2 = 1

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = 1 + 1

Reorder the terms:
1 + -2z + z2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2z + z2 = 2

Factor a perfect square on the left side:
(z + -1)(z + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1

z + -1 = 1.414213562 Simplifying z + -1 = 1.414213562 Reorder the terms: -1 + z = 1.414213562 Solving -1 + z = 1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + z = 1.414213562 + 1 z = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 z = 2.414213562 Simplifying z = 2.414213562

Subproblem 2

z + -1 = -1.414213562 Simplifying z + -1 = -1.414213562 Reorder the terms: -1 + z = -1.414213562 Solving -1 + z = -1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + z = -1.414213562 + 1 z = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 z = -0.414213562 Simplifying z = -0.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.414213562, -0.414213562}

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