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Simplifying 9z + 3(4z2 + -2) + -1(-3z + z + -2) = 0 Reorder the terms: 9z + 3(-2 + 4z2) + -1(-3z + z + -2) = 0 9z + (-2 * 3 + 4z2 * 3) + -1(-3z + z + -2) = 0 9z + (-6 + 12z2) + -1(-3z + z + -2) = 0 Reorder the terms: 9z + -6 + 12z2 + -1(-2 + -3z + z) = 0 Combine like terms: -3z + z = -2z 9z + -6 + 12z2 + -1(-2 + -2z) = 0 9z + -6 + 12z2 + (-2 * -1 + -2z * -1) = 0 9z + -6 + 12z2 + (2 + 2z) = 0 Reorder the terms: -6 + 2 + 9z + 2z + 12z2 = 0 Combine like terms: -6 + 2 = -4 -4 + 9z + 2z + 12z2 = 0 Combine like terms: 9z + 2z = 11z -4 + 11z + 12z2 = 0 Solving -4 + 11z + 12z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.3333333333 + 0.9166666667z + z2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.9166666667z + 0.3333333333 + z2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.9166666667z + z2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.9166666667z + z2 = 0 + 0.3333333333 0.9166666667z + z2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.9166666667z + z2 = 0.3333333333 The z term is 0.9166666667z. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667z + 0.2100694445 + z2 = 0.3333333333 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667z + z2 = 0.3333333333 + 0.2100694445 Combine like terms: 0.3333333333 + 0.2100694445 = 0.5434027778 0.2100694445 + 0.9166666667z + z2 = 0.5434027778 Factor a perfect square on the left side: (z + 0.4583333334)(z + 0.4583333334) = 0.5434027778 Calculate the square root of the right side: 0.737158584 Break this problem into two subproblems by setting (z + 0.4583333334) equal to 0.737158584 and -0.737158584.Subproblem 1
z + 0.4583333334 = 0.737158584 Simplifying z + 0.4583333334 = 0.737158584 Reorder the terms: 0.4583333334 + z = 0.737158584 Solving 0.4583333334 + z = 0.737158584 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + z = 0.737158584 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + z = 0.737158584 + -0.4583333334 z = 0.737158584 + -0.4583333334 Combine like terms: 0.737158584 + -0.4583333334 = 0.2788252506 z = 0.2788252506 Simplifying z = 0.2788252506Subproblem 2
z + 0.4583333334 = -0.737158584 Simplifying z + 0.4583333334 = -0.737158584 Reorder the terms: 0.4583333334 + z = -0.737158584 Solving 0.4583333334 + z = -0.737158584 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + z = -0.737158584 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + z = -0.737158584 + -0.4583333334 z = -0.737158584 + -0.4583333334 Combine like terms: -0.737158584 + -0.4583333334 = -1.1954919174 z = -1.1954919174 Simplifying z = -1.1954919174Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.2788252506, -1.1954919174}
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