(z^2)+z-28=0

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Solution for (z^2)+z-28=0 equation:


Simplifying
(z2) + z + -28 = 0
z2 + z + -28 = 0

Reorder the terms:
-28 + z + z2 = 0

Solving
-28 + z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + z + 28 + z2 = 0 + 28

Reorder the terms:
-28 + 28 + z + z2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + z + z2 = 0 + 28
z + z2 = 0 + 28

Combine like terms: 0 + 28 = 28
z + z2 = 28

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.
z + 0.25 + z2 = 28 + 0.25

Reorder the terms:
0.25 + z + z2 = 28 + 0.25

Combine like terms: 28 + 0.25 = 28.25
0.25 + z + z2 = 28.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 28.25

Calculate the square root of the right side: 5.315072906

Break this problem into two subproblems by setting 
(z + 0.5) equal to 5.315072906 and -5.315072906.

Subproblem 1

z + 0.5 = 5.315072906 Simplifying z + 0.5 = 5.315072906 Reorder the terms: 0.5 + z = 5.315072906 Solving 0.5 + z = 5.315072906 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 = 5.315072906 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 5.315072906 + -0.5 z = 5.315072906 + -0.5 Combine like terms: 5.315072906 + -0.5 = 4.815072906 z = 4.815072906 Simplifying z = 4.815072906

Subproblem 2

z + 0.5 = -5.315072906 Simplifying z + 0.5 = -5.315072906 Reorder the terms: 0.5 + z = -5.315072906 Solving 0.5 + z = -5.315072906 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 = -5.315072906 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -5.315072906 + -0.5 z = -5.315072906 + -0.5 Combine like terms: -5.315072906 + -0.5 = -5.815072906 z = -5.815072906 Simplifying z = -5.815072906

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.815072906, -5.815072906}

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