z^2+6z+7=0

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Solution for z^2+6z+7=0 equation:


Simplifying
z2 + 6z + 7 = 0

Reorder the terms:
7 + 6z + z2 = 0

Solving
7 + 6z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 6z + -7 + z2 = 0 + -7

Reorder the terms:
7 + -7 + 6z + z2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 6z + z2 = 0 + -7
6z + z2 = 0 + -7

Combine like terms: 0 + -7 = -7
6z + z2 = -7

The z term is 6z.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6z + 9 + z2 = -7 + 9

Reorder the terms:
9 + 6z + z2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + 6z + z2 = 2

Factor a perfect square on the left side:
(z + 3)(z + 3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z + 3) equal to 1.414213562 and -1.414213562.

Subproblem 1

z + 3 = 1.414213562 Simplifying z + 3 = 1.414213562 Reorder the terms: 3 + z = 1.414213562 Solving 3 + z = 1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + z = 1.414213562 + -3 z = 1.414213562 + -3 Combine like terms: 1.414213562 + -3 = -1.585786438 z = -1.585786438 Simplifying z = -1.585786438

Subproblem 2

z + 3 = -1.414213562 Simplifying z + 3 = -1.414213562 Reorder the terms: 3 + z = -1.414213562 Solving 3 + z = -1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + z = -1.414213562 + -3 z = -1.414213562 + -3 Combine like terms: -1.414213562 + -3 = -4.414213562 z = -4.414213562 Simplifying z = -4.414213562

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-1.585786438, -4.414213562}

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