z^2+4z=39

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Solution for z^2+4z=39 equation:


Simplifying
z2 + 4z = 39

Reorder the terms:
4z + z2 = 39

Solving
4z + z2 = 39

Solving for variable 'z'.

Reorder the terms:
-39 + 4z + z2 = 39 + -39

Combine like terms: 39 + -39 = 0
-39 + 4z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 4z + 39 + z2 = 0 + 39

Reorder the terms:
-39 + 39 + 4z + z2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 4z + z2 = 0 + 39
4z + z2 = 0 + 39

Combine like terms: 0 + 39 = 39
4z + z2 = 39

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = 39 + 4

Reorder the terms:
4 + 4z + z2 = 39 + 4

Combine like terms: 39 + 4 = 43
4 + 4z + z2 = 43

Factor a perfect square on the left side:
(z + 2)(z + 2) = 43

Calculate the square root of the right side: 6.557438524

Break this problem into two subproblems by setting 
(z + 2) equal to 6.557438524 and -6.557438524.

Subproblem 1

z + 2 = 6.557438524 Simplifying z + 2 = 6.557438524 Reorder the terms: 2 + z = 6.557438524 Solving 2 + z = 6.557438524 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 6.557438524 + -2 Combine like terms: 2 + -2 = 0 0 + z = 6.557438524 + -2 z = 6.557438524 + -2 Combine like terms: 6.557438524 + -2 = 4.557438524 z = 4.557438524 Simplifying z = 4.557438524

Subproblem 2

z + 2 = -6.557438524 Simplifying z + 2 = -6.557438524 Reorder the terms: 2 + z = -6.557438524 Solving 2 + z = -6.557438524 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -6.557438524 + -2 Combine like terms: 2 + -2 = 0 0 + z = -6.557438524 + -2 z = -6.557438524 + -2 Combine like terms: -6.557438524 + -2 = -8.557438524 z = -8.557438524 Simplifying z = -8.557438524

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.557438524, -8.557438524}

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