z^2+14z=-27

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Solution for z^2+14z=-27 equation:


Simplifying
z2 + 14z = -27

Reorder the terms:
14z + z2 = -27

Solving
14z + z2 = -27

Solving for variable 'z'.

Reorder the terms:
27 + 14z + z2 = -27 + 27

Combine like terms: -27 + 27 = 0
27 + 14z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-27' to each side of the equation.
27 + 14z + -27 + z2 = 0 + -27

Reorder the terms:
27 + -27 + 14z + z2 = 0 + -27

Combine like terms: 27 + -27 = 0
0 + 14z + z2 = 0 + -27
14z + z2 = 0 + -27

Combine like terms: 0 + -27 = -27
14z + z2 = -27

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 = -27 + 49

Reorder the terms:
49 + 14z + z2 = -27 + 49

Combine like terms: -27 + 49 = 22
49 + 14z + z2 = 22

Factor a perfect square on the left side:
(z + 7)(z + 7) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(z + 7) equal to 4.69041576 and -4.69041576.

Subproblem 1

z + 7 = 4.69041576 Simplifying z + 7 = 4.69041576 Reorder the terms: 7 + z = 4.69041576 Solving 7 + z = 4.69041576 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 = 4.69041576 + -7 Combine like terms: 7 + -7 = 0 0 + z = 4.69041576 + -7 z = 4.69041576 + -7 Combine like terms: 4.69041576 + -7 = -2.30958424 z = -2.30958424 Simplifying z = -2.30958424

Subproblem 2

z + 7 = -4.69041576 Simplifying z + 7 = -4.69041576 Reorder the terms: 7 + z = -4.69041576 Solving 7 + z = -4.69041576 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 = -4.69041576 + -7 Combine like terms: 7 + -7 = 0 0 + z = -4.69041576 + -7 z = -4.69041576 + -7 Combine like terms: -4.69041576 + -7 = -11.69041576 z = -11.69041576 Simplifying z = -11.69041576

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-2.30958424, -11.69041576}

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