z^2+16z+80=166

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Solution for z^2+16z+80=166 equation:


Simplifying
z2 + 16z + 80 = 166

Reorder the terms:
80 + 16z + z2 = 166

Solving
80 + 16z + z2 = 166

Solving for variable 'z'.

Reorder the terms:
80 + -166 + 16z + z2 = 166 + -166

Combine like terms: 80 + -166 = -86
-86 + 16z + z2 = 166 + -166

Combine like terms: 166 + -166 = 0
-86 + 16z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '86' to each side of the equation.
-86 + 16z + 86 + z2 = 0 + 86

Reorder the terms:
-86 + 86 + 16z + z2 = 0 + 86

Combine like terms: -86 + 86 = 0
0 + 16z + z2 = 0 + 86
16z + z2 = 0 + 86

Combine like terms: 0 + 86 = 86
16z + z2 = 86

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 = 86 + 64

Reorder the terms:
64 + 16z + z2 = 86 + 64

Combine like terms: 86 + 64 = 150
64 + 16z + z2 = 150

Factor a perfect square on the left side:
(z + 8)(z + 8) = 150

Calculate the square root of the right side: 12.247448714

Break this problem into two subproblems by setting 
(z + 8) equal to 12.247448714 and -12.247448714.

Subproblem 1

z + 8 = 12.247448714 Simplifying z + 8 = 12.247448714 Reorder the terms: 8 + z = 12.247448714 Solving 8 + z = 12.247448714 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 = 12.247448714 + -8 Combine like terms: 8 + -8 = 0 0 + z = 12.247448714 + -8 z = 12.247448714 + -8 Combine like terms: 12.247448714 + -8 = 4.247448714 z = 4.247448714 Simplifying z = 4.247448714

Subproblem 2

z + 8 = -12.247448714 Simplifying z + 8 = -12.247448714 Reorder the terms: 8 + z = -12.247448714 Solving 8 + z = -12.247448714 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 = -12.247448714 + -8 Combine like terms: 8 + -8 = 0 0 + z = -12.247448714 + -8 z = -12.247448714 + -8 Combine like terms: -12.247448714 + -8 = -20.247448714 z = -20.247448714 Simplifying z = -20.247448714

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.247448714, -20.247448714}

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