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Simplifying (-3z + 5 + 6z + -9) * 3z + -4 = 0 Reorder the terms: (5 + -9 + -3z + 6z) * 3z + -4 = 0 Combine like terms: 5 + -9 = -4 (-4 + -3z + 6z) * 3z + -4 = 0 Combine like terms: -3z + 6z = 3z (-4 + 3z) * 3z + -4 = 0 Reorder the terms for easier multiplication: 3z(-4 + 3z) + -4 = 0 (-4 * 3z + 3z * 3z) + -4 = 0 (-12z + 9z2) + -4 = 0 Reorder the terms: -4 + -12z + 9z2 = 0 Solving -4 + -12z + 9z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.4444444444 + -1.333333333z + z2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + -1.333333333z + 0.4444444444 + z2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + -1.333333333z + z2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + -1.333333333z + z2 = 0 + 0.4444444444 -1.333333333z + z2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 -1.333333333z + z2 = 0.4444444444 The z term is -1.333333333z. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333z + 0.4444444442 + z2 = 0.4444444444 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333z + z2 = 0.4444444444 + 0.4444444442 Combine like terms: 0.4444444444 + 0.4444444442 = 0.8888888886 0.4444444442 + -1.333333333z + z2 = 0.8888888886 Factor a perfect square on the left side: (z + -0.6666666665)(z + -0.6666666665) = 0.8888888886 Calculate the square root of the right side: 0.942809041 Break this problem into two subproblems by setting (z + -0.6666666665) equal to 0.942809041 and -0.942809041.Subproblem 1
z + -0.6666666665 = 0.942809041 Simplifying z + -0.6666666665 = 0.942809041 Reorder the terms: -0.6666666665 + z = 0.942809041 Solving -0.6666666665 + z = 0.942809041 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + z = 0.942809041 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + z = 0.942809041 + 0.6666666665 z = 0.942809041 + 0.6666666665 Combine like terms: 0.942809041 + 0.6666666665 = 1.6094757075 z = 1.6094757075 Simplifying z = 1.6094757075Subproblem 2
z + -0.6666666665 = -0.942809041 Simplifying z + -0.6666666665 = -0.942809041 Reorder the terms: -0.6666666665 + z = -0.942809041 Solving -0.6666666665 + z = -0.942809041 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + z = -0.942809041 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + z = -0.942809041 + 0.6666666665 z = -0.942809041 + 0.6666666665 Combine like terms: -0.942809041 + 0.6666666665 = -0.2761423745 z = -0.2761423745 Simplifying z = -0.2761423745Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.6094757075, -0.2761423745}
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