z^2+4=-6z

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Solution for z^2+4=-6z equation:


Simplifying
z2 + 4 = -6z

Reorder the terms:
4 + z2 = -6z

Solving
4 + z2 = -6z

Solving for variable 'z'.

Reorder the terms:
4 + 6z + z2 = -6z + 6z

Combine like terms: -6z + 6z = 0
4 + 6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 6z + -4 + z2 = 0 + -4

Reorder the terms:
4 + -4 + 6z + z2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 6z + z2 = 0 + -4
6z + z2 = 0 + -4

Combine like terms: 0 + -4 = -4
6z + z2 = -4

The z term is 6z.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6z + 9 + z2 = -4 + 9

Reorder the terms:
9 + 6z + z2 = -4 + 9

Combine like terms: -4 + 9 = 5
9 + 6z + z2 = 5

Factor a perfect square on the left side:
(z + 3)(z + 3) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(z + 3) equal to 2.236067978 and -2.236067978.

Subproblem 1

z + 3 = 2.236067978 Simplifying z + 3 = 2.236067978 Reorder the terms: 3 + z = 2.236067978 Solving 3 + z = 2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 2.236067978 + -3 Combine like terms: 3 + -3 = 0 0 + z = 2.236067978 + -3 z = 2.236067978 + -3 Combine like terms: 2.236067978 + -3 = -0.763932022 z = -0.763932022 Simplifying z = -0.763932022

Subproblem 2

z + 3 = -2.236067978 Simplifying z + 3 = -2.236067978 Reorder the terms: 3 + z = -2.236067978 Solving 3 + z = -2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -2.236067978 + -3 Combine like terms: 3 + -3 = 0 0 + z = -2.236067978 + -3 z = -2.236067978 + -3 Combine like terms: -2.236067978 + -3 = -5.236067978 z = -5.236067978 Simplifying z = -5.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.763932022, -5.236067978}

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