z^2+16z+52=0

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Solution for z^2+16z+52=0 equation:


Simplifying
z2 + 16z + 52 = 0

Reorder the terms:
52 + 16z + z2 = 0

Solving
52 + 16z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-52' to each side of the equation.
52 + 16z + -52 + z2 = 0 + -52

Reorder the terms:
52 + -52 + 16z + z2 = 0 + -52

Combine like terms: 52 + -52 = 0
0 + 16z + z2 = 0 + -52
16z + z2 = 0 + -52

Combine like terms: 0 + -52 = -52
16z + z2 = -52

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 = -52 + 64

Reorder the terms:
64 + 16z + z2 = -52 + 64

Combine like terms: -52 + 64 = 12
64 + 16z + z2 = 12

Factor a perfect square on the left side:
(z + 8)(z + 8) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(z + 8) equal to 3.464101615 and -3.464101615.

Subproblem 1

z + 8 = 3.464101615 Simplifying z + 8 = 3.464101615 Reorder the terms: 8 + z = 3.464101615 Solving 8 + z = 3.464101615 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 = 3.464101615 + -8 Combine like terms: 8 + -8 = 0 0 + z = 3.464101615 + -8 z = 3.464101615 + -8 Combine like terms: 3.464101615 + -8 = -4.535898385 z = -4.535898385 Simplifying z = -4.535898385

Subproblem 2

z + 8 = -3.464101615 Simplifying z + 8 = -3.464101615 Reorder the terms: 8 + z = -3.464101615 Solving 8 + z = -3.464101615 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 = -3.464101615 + -8 Combine like terms: 8 + -8 = 0 0 + z = -3.464101615 + -8 z = -3.464101615 + -8 Combine like terms: -3.464101615 + -8 = -11.464101615 z = -11.464101615 Simplifying z = -11.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-4.535898385, -11.464101615}

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