6z^2+7z-2=0

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Solution for 6z^2+7z-2=0 equation:


Simplifying
6z2 + 7z + -2 = 0

Reorder the terms:
-2 + 7z + 6z2 = 0

Solving
-2 + 7z + 6z2 = 0

Solving for variable 'z'.

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

Divide each side by '6'.
-0.3333333333 + 1.166666667z + z2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 1.166666667z + 0.3333333333 + z2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 1.166666667z + z2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 1.166666667z + z2 = 0 + 0.3333333333
1.166666667z + z2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
1.166666667z + z2 = 0.3333333333

The z term is 1.166666667z.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667z + 0.3402777780 + z2 = 0.3333333333 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667z + z2 = 0.3333333333 + 0.3402777780

Combine like terms: 0.3333333333 + 0.3402777780 = 0.6736111113
0.3402777780 + 1.166666667z + z2 = 0.6736111113

Factor a perfect square on the left side:
(z + 0.5833333335)(z + 0.5833333335) = 0.6736111113

Calculate the square root of the right side: 0.82073815

Break this problem into two subproblems by setting 
(z + 0.5833333335) equal to 0.82073815 and -0.82073815.

Subproblem 1

z + 0.5833333335 = 0.82073815 Simplifying z + 0.5833333335 = 0.82073815 Reorder the terms: 0.5833333335 + z = 0.82073815 Solving 0.5833333335 + z = 0.82073815 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + z = 0.82073815 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + z = 0.82073815 + -0.5833333335 z = 0.82073815 + -0.5833333335 Combine like terms: 0.82073815 + -0.5833333335 = 0.2374048165 z = 0.2374048165 Simplifying z = 0.2374048165

Subproblem 2

z + 0.5833333335 = -0.82073815 Simplifying z + 0.5833333335 = -0.82073815 Reorder the terms: 0.5833333335 + z = -0.82073815 Solving 0.5833333335 + z = -0.82073815 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + z = -0.82073815 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + z = -0.82073815 + -0.5833333335 z = -0.82073815 + -0.5833333335 Combine like terms: -0.82073815 + -0.5833333335 = -1.4040714835 z = -1.4040714835 Simplifying z = -1.4040714835

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.2374048165, -1.4040714835}

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