z^2+16z=-5

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Solution for z^2+16z=-5 equation:


Simplifying
z2 + 16z = -5

Reorder the terms:
16z + z2 = -5

Solving
16z + z2 = -5

Solving for variable 'z'.

Reorder the terms:
5 + 16z + z2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + 16z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 16z + -5 + z2 = 0 + -5

Reorder the terms:
5 + -5 + 16z + z2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 16z + z2 = 0 + -5
16z + z2 = 0 + -5

Combine like terms: 0 + -5 = -5
16z + z2 = -5

The z term is 16z.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16z + 64 + z2 = -5 + 64

Reorder the terms:
64 + 16z + z2 = -5 + 64

Combine like terms: -5 + 64 = 59
64 + 16z + z2 = 59

Factor a perfect square on the left side:
(z + 8)(z + 8) = 59

Calculate the square root of the right side: 7.681145748

Break this problem into two subproblems by setting 
(z + 8) equal to 7.681145748 and -7.681145748.

Subproblem 1

z + 8 = 7.681145748 Simplifying z + 8 = 7.681145748 Reorder the terms: 8 + z = 7.681145748 Solving 8 + z = 7.681145748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = 7.681145748 + -8 Combine like terms: 8 + -8 = 0 0 + z = 7.681145748 + -8 z = 7.681145748 + -8 Combine like terms: 7.681145748 + -8 = -0.318854252 z = -0.318854252 Simplifying z = -0.318854252

Subproblem 2

z + 8 = -7.681145748 Simplifying z + 8 = -7.681145748 Reorder the terms: 8 + z = -7.681145748 Solving 8 + z = -7.681145748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + z = -7.681145748 + -8 Combine like terms: 8 + -8 = 0 0 + z = -7.681145748 + -8 z = -7.681145748 + -8 Combine like terms: -7.681145748 + -8 = -15.681145748 z = -15.681145748 Simplifying z = -15.681145748

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.318854252, -15.681145748}

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