z^2+7z+5=0

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Solution for z^2+7z+5=0 equation:


Simplifying
z2 + 7z + 5 = 0

Reorder the terms:
5 + 7z + z2 = 0

Solving
5 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 7z + -5 + z2 = 0 + -5

Reorder the terms:
5 + -5 + 7z + z2 = 0 + -5

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

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

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 = -5 + 12.25

Reorder the terms:
12.25 + 7z + z2 = -5 + 12.25

Combine like terms: -5 + 12.25 = 7.25
12.25 + 7z + z2 = 7.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(z + 3.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

z + 3.5 = 2.692582404 Simplifying z + 3.5 = 2.692582404 Reorder the terms: 3.5 + z = 2.692582404 Solving 3.5 + z = 2.692582404 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 = 2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 2.692582404 + -3.5 z = 2.692582404 + -3.5 Combine like terms: 2.692582404 + -3.5 = -0.807417596 z = -0.807417596 Simplifying z = -0.807417596

Subproblem 2

z + 3.5 = -2.692582404 Simplifying z + 3.5 = -2.692582404 Reorder the terms: 3.5 + z = -2.692582404 Solving 3.5 + z = -2.692582404 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 = -2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -2.692582404 + -3.5 z = -2.692582404 + -3.5 Combine like terms: -2.692582404 + -3.5 = -6.192582404 z = -6.192582404 Simplifying z = -6.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.807417596, -6.192582404}

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