2z^2+11z+6=0

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Solution for 2z^2+11z+6=0 equation:


Simplifying
2z2 + 11z + 6 = 0

Reorder the terms:
6 + 11z + 2z2 = 0

Solving
6 + 11z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
3 + 5.5z + z2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5.5z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 5.5z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5.5z + z2 = 0 + -3
5.5z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
5.5z + z2 = -3

The z term is 5.5z.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5z + 7.5625 + z2 = -3 + 7.5625

Reorder the terms:
7.5625 + 5.5z + z2 = -3 + 7.5625

Combine like terms: -3 + 7.5625 = 4.5625
7.5625 + 5.5z + z2 = 4.5625

Factor a perfect square on the left side:
(z + 2.75)(z + 2.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(z + 2.75) equal to 2.136000936 and -2.136000936.

Subproblem 1

z + 2.75 = 2.136000936 Simplifying z + 2.75 = 2.136000936 Reorder the terms: 2.75 + z = 2.136000936 Solving 2.75 + z = 2.136000936 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + z = 2.136000936 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + z = 2.136000936 + -2.75 z = 2.136000936 + -2.75 Combine like terms: 2.136000936 + -2.75 = -0.613999064 z = -0.613999064 Simplifying z = -0.613999064

Subproblem 2

z + 2.75 = -2.136000936 Simplifying z + 2.75 = -2.136000936 Reorder the terms: 2.75 + z = -2.136000936 Solving 2.75 + z = -2.136000936 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + z = -2.136000936 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + z = -2.136000936 + -2.75 z = -2.136000936 + -2.75 Combine like terms: -2.136000936 + -2.75 = -4.886000936 z = -4.886000936 Simplifying z = -4.886000936

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.613999064, -4.886000936}

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