18z^2+37z+16=0

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Solution for 18z^2+37z+16=0 equation:


Simplifying
18z2 + 37z + 16 = 0

Reorder the terms:
16 + 37z + 18z2 = 0

Solving
16 + 37z + 18z2 = 0

Solving for variable 'z'.

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

Divide each side by '18'.
0.8888888889 + 2.055555556z + z2 = 0

Move the constant term to the right:

Add '-0.8888888889' to each side of the equation.
0.8888888889 + 2.055555556z + -0.8888888889 + z2 = 0 + -0.8888888889

Reorder the terms:
0.8888888889 + -0.8888888889 + 2.055555556z + z2 = 0 + -0.8888888889

Combine like terms: 0.8888888889 + -0.8888888889 = 0.0000000000
0.0000000000 + 2.055555556z + z2 = 0 + -0.8888888889
2.055555556z + z2 = 0 + -0.8888888889

Combine like terms: 0 + -0.8888888889 = -0.8888888889
2.055555556z + z2 = -0.8888888889

The z term is 2.055555556z.  Take half its coefficient (1.027777778).
Square it (1.056327161) and add it to both sides.

Add '1.056327161' to each side of the equation.
2.055555556z + 1.056327161 + z2 = -0.8888888889 + 1.056327161

Reorder the terms:
1.056327161 + 2.055555556z + z2 = -0.8888888889 + 1.056327161

Combine like terms: -0.8888888889 + 1.056327161 = 0.1674382721
1.056327161 + 2.055555556z + z2 = 0.1674382721

Factor a perfect square on the left side:
(z + 1.027777778)(z + 1.027777778) = 0.1674382721

Calculate the square root of the right side: 0.409192219

Break this problem into two subproblems by setting 
(z + 1.027777778) equal to 0.409192219 and -0.409192219.

Subproblem 1

z + 1.027777778 = 0.409192219 Simplifying z + 1.027777778 = 0.409192219 Reorder the terms: 1.027777778 + z = 0.409192219 Solving 1.027777778 + z = 0.409192219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.027777778' to each side of the equation. 1.027777778 + -1.027777778 + z = 0.409192219 + -1.027777778 Combine like terms: 1.027777778 + -1.027777778 = 0.000000000 0.000000000 + z = 0.409192219 + -1.027777778 z = 0.409192219 + -1.027777778 Combine like terms: 0.409192219 + -1.027777778 = -0.618585559 z = -0.618585559 Simplifying z = -0.618585559

Subproblem 2

z + 1.027777778 = -0.409192219 Simplifying z + 1.027777778 = -0.409192219 Reorder the terms: 1.027777778 + z = -0.409192219 Solving 1.027777778 + z = -0.409192219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.027777778' to each side of the equation. 1.027777778 + -1.027777778 + z = -0.409192219 + -1.027777778 Combine like terms: 1.027777778 + -1.027777778 = 0.000000000 0.000000000 + z = -0.409192219 + -1.027777778 z = -0.409192219 + -1.027777778 Combine like terms: -0.409192219 + -1.027777778 = -1.436969997 z = -1.436969997 Simplifying z = -1.436969997

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.618585559, -1.436969997}

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