z^2+4z=23

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Solution for z^2+4z=23 equation:


Simplifying
z2 + 4z = 23

Reorder the terms:
4z + z2 = 23

Solving
4z + z2 = 23

Solving for variable 'z'.

Reorder the terms:
-23 + 4z + z2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 4z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 4z + 23 + z2 = 0 + 23

Reorder the terms:
-23 + 23 + 4z + z2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 4z + z2 = 0 + 23
4z + z2 = 0 + 23

Combine like terms: 0 + 23 = 23
4z + z2 = 23

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 = 23 + 4

Reorder the terms:
4 + 4z + z2 = 23 + 4

Combine like terms: 23 + 4 = 27
4 + 4z + z2 = 27

Factor a perfect square on the left side:
(z + 2)(z + 2) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(z + 2) equal to 5.196152423 and -5.196152423.

Subproblem 1

z + 2 = 5.196152423 Simplifying z + 2 = 5.196152423 Reorder the terms: 2 + z = 5.196152423 Solving 2 + z = 5.196152423 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 = 5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + z = 5.196152423 + -2 z = 5.196152423 + -2 Combine like terms: 5.196152423 + -2 = 3.196152423 z = 3.196152423 Simplifying z = 3.196152423

Subproblem 2

z + 2 = -5.196152423 Simplifying z + 2 = -5.196152423 Reorder the terms: 2 + z = -5.196152423 Solving 2 + z = -5.196152423 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 = -5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + z = -5.196152423 + -2 z = -5.196152423 + -2 Combine like terms: -5.196152423 + -2 = -7.196152423 z = -7.196152423 Simplifying z = -7.196152423

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.196152423, -7.196152423}

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