z^2+2z+2=7

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Solution for z^2+2z+2=7 equation:


Simplifying
z2 + 2z + 2 = 7

Reorder the terms:
2 + 2z + z2 = 7

Solving
2 + 2z + z2 = 7

Solving for variable 'z'.

Reorder the terms:
2 + -7 + 2z + z2 = 7 + -7

Combine like terms: 2 + -7 = -5
-5 + 2z + z2 = 7 + -7

Combine like terms: 7 + -7 = 0
-5 + 2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 2z + 5 + z2 = 0 + 5

Reorder the terms:
-5 + 5 + 2z + z2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 2z + z2 = 0 + 5
2z + z2 = 0 + 5

Combine like terms: 0 + 5 = 5
2z + z2 = 5

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 = 5 + 1

Reorder the terms:
1 + 2z + z2 = 5 + 1

Combine like terms: 5 + 1 = 6
1 + 2z + z2 = 6

Factor a perfect square on the left side:
(z + 1)(z + 1) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(z + 1) equal to 2.449489743 and -2.449489743.

Subproblem 1

z + 1 = 2.449489743 Simplifying z + 1 = 2.449489743 Reorder the terms: 1 + z = 2.449489743 Solving 1 + z = 2.449489743 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 = 2.449489743 + -1 Combine like terms: 1 + -1 = 0 0 + z = 2.449489743 + -1 z = 2.449489743 + -1 Combine like terms: 2.449489743 + -1 = 1.449489743 z = 1.449489743 Simplifying z = 1.449489743

Subproblem 2

z + 1 = -2.449489743 Simplifying z + 1 = -2.449489743 Reorder the terms: 1 + z = -2.449489743 Solving 1 + z = -2.449489743 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 = -2.449489743 + -1 Combine like terms: 1 + -1 = 0 0 + z = -2.449489743 + -1 z = -2.449489743 + -1 Combine like terms: -2.449489743 + -1 = -3.449489743 z = -3.449489743 Simplifying z = -3.449489743

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.449489743, -3.449489743}

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