z^2+12z+36=-2z

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Solution for z^2+12z+36=-2z equation:


Simplifying
z2 + 12z + 36 = -2z

Reorder the terms:
36 + 12z + z2 = -2z

Solving
36 + 12z + z2 = -2z

Solving for variable 'z'.

Reorder the terms:
36 + 12z + 2z + z2 = -2z + 2z

Combine like terms: 12z + 2z = 14z
36 + 14z + z2 = -2z + 2z

Combine like terms: -2z + 2z = 0
36 + 14z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-36' to each side of the equation.
36 + 14z + -36 + z2 = 0 + -36

Reorder the terms:
36 + -36 + 14z + z2 = 0 + -36

Combine like terms: 36 + -36 = 0
0 + 14z + z2 = 0 + -36
14z + z2 = 0 + -36

Combine like terms: 0 + -36 = -36
14z + z2 = -36

The z term is 14z.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14z + 49 + z2 = -36 + 49

Reorder the terms:
49 + 14z + z2 = -36 + 49

Combine like terms: -36 + 49 = 13
49 + 14z + z2 = 13

Factor a perfect square on the left side:
(z + 7)(z + 7) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(z + 7) equal to 3.605551275 and -3.605551275.

Subproblem 1

z + 7 = 3.605551275 Simplifying z + 7 = 3.605551275 Reorder the terms: 7 + z = 3.605551275 Solving 7 + z = 3.605551275 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = 3.605551275 + -7 Combine like terms: 7 + -7 = 0 0 + z = 3.605551275 + -7 z = 3.605551275 + -7 Combine like terms: 3.605551275 + -7 = -3.394448725 z = -3.394448725 Simplifying z = -3.394448725

Subproblem 2

z + 7 = -3.605551275 Simplifying z + 7 = -3.605551275 Reorder the terms: 7 + z = -3.605551275 Solving 7 + z = -3.605551275 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = -3.605551275 + -7 Combine like terms: 7 + -7 = 0 0 + z = -3.605551275 + -7 z = -3.605551275 + -7 Combine like terms: -3.605551275 + -7 = -10.605551275 z = -10.605551275 Simplifying z = -10.605551275

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-3.394448725, -10.605551275}

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