z^2+312.8z=37223

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Solution for z^2+312.8z=37223 equation: 


Simplifying
z2 + 312.8z = 37223

Reorder the terms:
312.8z + z2 = 37223

Solving
312.8z + z2 = 37223

Solving for variable 'z'.

Reorder the terms:
-37223 + 312.8z + z2 = 37223 + -37223

Combine like terms: 37223 + -37223 = 0
-37223 + 312.8z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '37223' to each side of the equation.
-37223 + 312.8z + 37223 + z2 = 0 + 37223

Reorder the terms:
-37223 + 37223 + 312.8z + z2 = 0 + 37223

Combine like terms: -37223 + 37223 = 0
0 + 312.8z + z2 = 0 + 37223
312.8z + z2 = 0 + 37223

Combine like terms: 0 + 37223 = 37223
312.8z + z2 = 37223

The z term is 312.8z.  Take half its coefficient (156.4).
Square it (24460.96) and add it to both sides.

Add '24460.96' to each side of the equation.
312.8z + 24460.96 + z2 = 37223 + 24460.96

Reorder the terms:
24460.96 + 312.8z + z2 = 37223 + 24460.96

Combine like terms: 37223 + 24460.96 = 61683.96
24460.96 + 312.8z + z2 = 61683.96

Factor a perfect square on the left side:
(z + 156.4)(z + 156.4) = 61683.96

Calculate the square root of the right side: 248.362557565

Break this problem into two subproblems by setting 
(z + 156.4) equal to 248.362557565 and -248.362557565.


Subproblem 1
z + 156.4 = 248.362557565

Simplifying
z + 156.4 = 248.362557565

Reorder the terms:
156.4 + z = 248.362557565

Solving
156.4 + z = 248.362557565

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-156.4' to each side of the equation.
156.4 + -156.4 + z = 248.362557565 + -156.4

Combine like terms: 156.4 + -156.4 = 0.0
0.0 + z = 248.362557565 + -156.4
z = 248.362557565 + -156.4

Combine like terms: 248.362557565 + -156.4 = 91.962557565
z = 91.962557565

Simplifying
z = 91.962557565


Subproblem 2
z + 156.4 = -248.362557565

Simplifying
z + 156.4 = -248.362557565

Reorder the terms:
156.4 + z = -248.362557565

Solving
156.4 + z = -248.362557565

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-156.4' to each side of the equation.
156.4 + -156.4 + z = -248.362557565 + -156.4

Combine like terms: 156.4 + -156.4 = 0.0
0.0 + z = -248.362557565 + -156.4
z = -248.362557565 + -156.4

Combine like terms: -248.362557565 + -156.4 = -404.762557565
z = -404.762557565

Simplifying
z = -404.762557565


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {91.962557565, -404.762557565}

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