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