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Simplifying -1(3z2 + 4z) + -1(6z2 + -2) = 0 Reorder the terms: -1(4z + 3z2) + -1(6z2 + -2) = 0 (4z * -1 + 3z2 * -1) + -1(6z2 + -2) = 0 (-4z + -3z2) + -1(6z2 + -2) = 0 Reorder the terms: -4z + -3z2 + -1(-2 + 6z2) = 0 -4z + -3z2 + (-2 * -1 + 6z2 * -1) = 0 -4z + -3z2 + (2 + -6z2) = 0 Reorder the terms: 2 + -4z + -3z2 + -6z2 = 0 Combine like terms: -3z2 + -6z2 = -9z2 2 + -4z + -9z2 = 0 Solving 2 + -4z + -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'. -0.2222222222 + 0.4444444444z + z2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.4444444444z + 0.2222222222 + z2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + 0.4444444444z + z2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + 0.4444444444z + z2 = 0 + 0.2222222222 0.4444444444z + z2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 0.4444444444z + z2 = 0.2222222222 The z term is 0.4444444444z. Take half its coefficient (0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. 0.4444444444z + 0.04938271604 + z2 = 0.2222222222 + 0.04938271604 Reorder the terms: 0.04938271604 + 0.4444444444z + z2 = 0.2222222222 + 0.04938271604 Combine like terms: 0.2222222222 + 0.04938271604 = 0.27160493824 0.04938271604 + 0.4444444444z + z2 = 0.27160493824 Factor a perfect square on the left side: (z + 0.2222222222)(z + 0.2222222222) = 0.27160493824 Calculate the square root of the right side: 0.521157307 Break this problem into two subproblems by setting (z + 0.2222222222) equal to 0.521157307 and -0.521157307.Subproblem 1
z + 0.2222222222 = 0.521157307 Simplifying z + 0.2222222222 = 0.521157307 Reorder the terms: 0.2222222222 + z = 0.521157307 Solving 0.2222222222 + z = 0.521157307 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + z = 0.521157307 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + z = 0.521157307 + -0.2222222222 z = 0.521157307 + -0.2222222222 Combine like terms: 0.521157307 + -0.2222222222 = 0.2989350848 z = 0.2989350848 Simplifying z = 0.2989350848Subproblem 2
z + 0.2222222222 = -0.521157307 Simplifying z + 0.2222222222 = -0.521157307 Reorder the terms: 0.2222222222 + z = -0.521157307 Solving 0.2222222222 + z = -0.521157307 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + z = -0.521157307 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + z = -0.521157307 + -0.2222222222 z = -0.521157307 + -0.2222222222 Combine like terms: -0.521157307 + -0.2222222222 = -0.7433795292 z = -0.7433795292 Simplifying z = -0.7433795292Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.2989350848, -0.7433795292}
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