2z^2+312.8z=37223

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


Simplifying
2z2 + 312.8z = 37223

Reorder the terms:
312.8z + 2z2 = 37223

Solving
312.8z + 2z2 = 37223

Solving for variable 'z'.

Reorder the terms:
-37223 + 312.8z + 2z2 = 37223 + -37223

Combine like terms: 37223 + -37223 = 0
-37223 + 312.8z + 2z2 = 0

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

Divide each side by '2'.
-18611.5 + 156.4z + z2 = 0

Move the constant term to the right:

Add '18611.5' to each side of the equation.
-18611.5 + 156.4z + 18611.5 + z2 = 0 + 18611.5

Reorder the terms:
-18611.5 + 18611.5 + 156.4z + z2 = 0 + 18611.5

Combine like terms: -18611.5 + 18611.5 = 0.0
0.0 + 156.4z + z2 = 0 + 18611.5
156.4z + z2 = 0 + 18611.5

Combine like terms: 0 + 18611.5 = 18611.5
156.4z + z2 = 18611.5

The z term is 156.4z.  Take half its coefficient (78.2).
Square it (6115.24) and add it to both sides.

Add '6115.24' to each side of the equation.
156.4z + 6115.24 + z2 = 18611.5 + 6115.24

Reorder the terms:
6115.24 + 156.4z + z2 = 18611.5 + 6115.24

Combine like terms: 18611.5 + 6115.24 = 24726.74
6115.24 + 156.4z + z2 = 24726.74

Factor a perfect square on the left side:
(z + 78.2)(z + 78.2) = 24726.74

Calculate the square root of the right side: 157.247384716

Break this problem into two subproblems by setting 
(z + 78.2) equal to 157.247384716 and -157.247384716.

Subproblem 1

z + 78.2 = 157.247384716 Simplifying z + 78.2 = 157.247384716 Reorder the terms: 78.2 + z = 157.247384716 Solving 78.2 + z = 157.247384716 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-78.2' to each side of the equation. 78.2 + -78.2 + z = 157.247384716 + -78.2 Combine like terms: 78.2 + -78.2 = 0.0 0.0 + z = 157.247384716 + -78.2 z = 157.247384716 + -78.2 Combine like terms: 157.247384716 + -78.2 = 79.047384716 z = 79.047384716 Simplifying z = 79.047384716

Subproblem 2

z + 78.2 = -157.247384716 Simplifying z + 78.2 = -157.247384716 Reorder the terms: 78.2 + z = -157.247384716 Solving 78.2 + z = -157.247384716 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-78.2' to each side of the equation. 78.2 + -78.2 + z = -157.247384716 + -78.2 Combine like terms: 78.2 + -78.2 = 0.0 0.0 + z = -157.247384716 + -78.2 z = -157.247384716 + -78.2 Combine like terms: -157.247384716 + -78.2 = -235.447384716 z = -235.447384716 Simplifying z = -235.447384716

Solution

The solution to the problem is based on the solutions from the subproblems. z = {79.047384716, -235.447384716}

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