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Simplifying 6z2 + -11z + -10 = 1 Reorder the terms: -10 + -11z + 6z2 = 1 Solving -10 + -11z + 6z2 = 1 Solving for variable 'z'. Reorder the terms: -10 + -1 + -11z + 6z2 = 1 + -1 Combine like terms: -10 + -1 = -11 -11 + -11z + 6z2 = 1 + -1 Combine like terms: 1 + -1 = 0 -11 + -11z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.833333333 + -1.833333333z + z2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + -1.833333333z + 1.833333333 + z2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + -1.833333333z + z2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + -1.833333333z + z2 = 0 + 1.833333333 -1.833333333z + z2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 -1.833333333z + z2 = 1.833333333 The z term is -1.833333333z. Take half its coefficient (-0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. -1.833333333z + 0.8402777775 + z2 = 1.833333333 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333z + z2 = 1.833333333 + 0.8402777775 Combine like terms: 1.833333333 + 0.8402777775 = 2.6736111105 0.8402777775 + -1.833333333z + z2 = 2.6736111105 Factor a perfect square on the left side: (z + -0.9166666665)(z + -0.9166666665) = 2.6736111105 Calculate the square root of the right side: 1.635118072 Break this problem into two subproblems by setting (z + -0.9166666665) equal to 1.635118072 and -1.635118072.Subproblem 1
z + -0.9166666665 = 1.635118072 Simplifying z + -0.9166666665 = 1.635118072 Reorder the terms: -0.9166666665 + z = 1.635118072 Solving -0.9166666665 + z = 1.635118072 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + z = 1.635118072 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + z = 1.635118072 + 0.9166666665 z = 1.635118072 + 0.9166666665 Combine like terms: 1.635118072 + 0.9166666665 = 2.5517847385 z = 2.5517847385 Simplifying z = 2.5517847385Subproblem 2
z + -0.9166666665 = -1.635118072 Simplifying z + -0.9166666665 = -1.635118072 Reorder the terms: -0.9166666665 + z = -1.635118072 Solving -0.9166666665 + z = -1.635118072 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + z = -1.635118072 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + z = -1.635118072 + 0.9166666665 z = -1.635118072 + 0.9166666665 Combine like terms: -1.635118072 + 0.9166666665 = -0.7184514055 z = -0.7184514055 Simplifying z = -0.7184514055Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.5517847385, -0.7184514055}
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