-z^2+z+13=0

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Solution for -z^2+z+13=0 equation:


Simplifying
-1z2 + z + 13 = 0

Reorder the terms:
13 + z + -1z2 = 0

Solving
13 + z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-13 + -1z + z2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -1z + 13 + z2 = 0 + 13

Reorder the terms:
-13 + 13 + -1z + z2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -1z + z2 = 0 + 13
-1z + z2 = 0 + 13

Combine like terms: 0 + 13 = 13
-1z + z2 = 13

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 13 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 13 + 0.25

Combine like terms: 13 + 0.25 = 13.25
0.25 + -1z + z2 = 13.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(z + 0.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

z + 0.5 = 3.640054945 Simplifying z + 0.5 = 3.640054945 Reorder the terms: 0.5 + z = 3.640054945 Solving 0.5 + z = 3.640054945 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 3.640054945 + -0.5 z = 3.640054945 + -0.5 Combine like terms: 3.640054945 + -0.5 = 3.140054945 z = 3.140054945 Simplifying z = 3.140054945

Subproblem 2

z + 0.5 = -3.640054945 Simplifying z + 0.5 = -3.640054945 Reorder the terms: 0.5 + z = -3.640054945 Solving 0.5 + z = -3.640054945 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -3.640054945 + -0.5 z = -3.640054945 + -0.5 Combine like terms: -3.640054945 + -0.5 = -4.140054945 z = -4.140054945 Simplifying z = -4.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.140054945, -4.140054945}

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