z^2-4z-23=0

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Solution for z^2-4z-23=0 equation:


Simplifying
z2 + -4z + -23 = 0

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

Solving
-23 + -4z + z2 = 0

Solving for variable 'z'.

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 = 7.196152423 z = 7.196152423 Simplifying z = 7.196152423

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 = -3.196152423 z = -3.196152423 Simplifying z = -3.196152423

Solution

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

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