z^2+7z+1=0

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Solution for z^2+7z+1=0 equation:


Simplifying
z2 + 7z + 1 = 0

Reorder the terms:
1 + 7z + z2 = 0

Solving
1 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 7z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 7z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 7z + z2 = 0 + -1
7z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
7z + z2 = -1

The z term is 7z.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7z + 12.25 + z2 = -1 + 12.25

Reorder the terms:
12.25 + 7z + z2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + 7z + z2 = 11.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(z + 3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

z + 3.5 = 3.354101966 Simplifying z + 3.5 = 3.354101966 Reorder the terms: 3.5 + z = 3.354101966 Solving 3.5 + z = 3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 3.354101966 + -3.5 z = 3.354101966 + -3.5 Combine like terms: 3.354101966 + -3.5 = -0.145898034 z = -0.145898034 Simplifying z = -0.145898034

Subproblem 2

z + 3.5 = -3.354101966 Simplifying z + 3.5 = -3.354101966 Reorder the terms: 3.5 + z = -3.354101966 Solving 3.5 + z = -3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -3.354101966 + -3.5 z = -3.354101966 + -3.5 Combine like terms: -3.354101966 + -3.5 = -6.854101966 z = -6.854101966 Simplifying z = -6.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.145898034, -6.854101966}

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