3z^2-11z-6=0

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Solution for 3z^2-11z-6=0 equation:


Simplifying
3z2 + -11z + -6 = 0

Reorder the terms:
-6 + -11z + 3z2 = 0

Solving
-6 + -11z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
-2 + -3.666666667z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -3.666666667z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -3.666666667z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -3.666666667z + z2 = 0 + 2
-3.666666667z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
-3.666666667z + z2 = 2

The z term is -3.666666667z.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667z + 3.361111114 + z2 = 2 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667z + z2 = 2 + 3.361111114

Combine like terms: 2 + 3.361111114 = 5.361111114
3.361111114 + -3.666666667z + z2 = 5.361111114

Factor a perfect square on the left side:
(z + -1.833333334)(z + -1.833333334) = 5.361111114

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(z + -1.833333334) equal to 2.315407332 and -2.315407332.

Subproblem 1

z + -1.833333334 = 2.315407332 Simplifying z + -1.833333334 = 2.315407332 Reorder the terms: -1.833333334 + z = 2.315407332 Solving -1.833333334 + z = 2.315407332 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + z = 2.315407332 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + z = 2.315407332 + 1.833333334 z = 2.315407332 + 1.833333334 Combine like terms: 2.315407332 + 1.833333334 = 4.148740666 z = 4.148740666 Simplifying z = 4.148740666

Subproblem 2

z + -1.833333334 = -2.315407332 Simplifying z + -1.833333334 = -2.315407332 Reorder the terms: -1.833333334 + z = -2.315407332 Solving -1.833333334 + z = -2.315407332 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + z = -2.315407332 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + z = -2.315407332 + 1.833333334 z = -2.315407332 + 1.833333334 Combine like terms: -2.315407332 + 1.833333334 = -0.482073998 z = -0.482073998 Simplifying z = -0.482073998

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.148740666, -0.482073998}

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