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Simplifying -4(2z + 9) = 3z(1 + -2z) Reorder the terms: -4(9 + 2z) = 3z(1 + -2z) (9 * -4 + 2z * -4) = 3z(1 + -2z) (-36 + -8z) = 3z(1 + -2z) -36 + -8z = (1 * 3z + -2z * 3z) -36 + -8z = (3z + -6z2) Solving -36 + -8z = 3z + -6z2 Solving for variable 'z'. Combine like terms: -8z + -3z = -11z -36 + -11z + 6z2 = 3z + -6z2 + -3z + 6z2 Reorder the terms: -36 + -11z + 6z2 = 3z + -3z + -6z2 + 6z2 Combine like terms: 3z + -3z = 0 -36 + -11z + 6z2 = 0 + -6z2 + 6z2 -36 + -11z + 6z2 = -6z2 + 6z2 Combine like terms: -6z2 + 6z2 = 0 -36 + -11z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -6 + -1.833333333z + z2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -1.833333333z + 6 + z2 = 0 + 6 Reorder the terms: -6 + 6 + -1.833333333z + z2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -1.833333333z + z2 = 0 + 6 -1.833333333z + z2 = 0 + 6 Combine like terms: 0 + 6 = 6 -1.833333333z + z2 = 6 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 = 6 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333z + z2 = 6 + 0.8402777775 Combine like terms: 6 + 0.8402777775 = 6.8402777775 0.8402777775 + -1.833333333z + z2 = 6.8402777775 Factor a perfect square on the left side: (z + -0.9166666665)(z + -0.9166666665) = 6.8402777775 Calculate the square root of the right side: 2.615392471 Break this problem into two subproblems by setting (z + -0.9166666665) equal to 2.615392471 and -2.615392471.Subproblem 1
z + -0.9166666665 = 2.615392471 Simplifying z + -0.9166666665 = 2.615392471 Reorder the terms: -0.9166666665 + z = 2.615392471 Solving -0.9166666665 + z = 2.615392471 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 = 2.615392471 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + z = 2.615392471 + 0.9166666665 z = 2.615392471 + 0.9166666665 Combine like terms: 2.615392471 + 0.9166666665 = 3.5320591375 z = 3.5320591375 Simplifying z = 3.5320591375Subproblem 2
z + -0.9166666665 = -2.615392471 Simplifying z + -0.9166666665 = -2.615392471 Reorder the terms: -0.9166666665 + z = -2.615392471 Solving -0.9166666665 + z = -2.615392471 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 = -2.615392471 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + z = -2.615392471 + 0.9166666665 z = -2.615392471 + 0.9166666665 Combine like terms: -2.615392471 + 0.9166666665 = -1.6987258045 z = -1.6987258045 Simplifying z = -1.6987258045Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.5320591375, -1.6987258045}
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